Basics of Real Analysis ...or “the weirdness of infinity,” day 2

Andrew Lin January 27, 2021 Final announcements

• Everything from Monday is still true (pset, course evals, office hours) • Final grades • Join the Course 18 Piazza and Discord!

1 Some history

Ancient Greeks complained about a lot of mathematics:

Example (Zeno’s paradox) Suppose we wanted to walk down the Infinite Corridor. To do this, we have to first walk down the first half of the path, but before that, we need to first walk the first quarter of the path, and before that, the first eighth, and so on. There’s always an “earlier” action to take, and “it’s impossible to do infinitely many actions” (says Zeno).

The usual explanation is that we can add up 1 1 1 + + + ··· = 1. 2 4 8 It’s a bit hard to convince the Greeks why this resolves the paradox, though...

2 Some history

Ancient Greeks complained about a lot of mathematics:

Example (Zeno’s paradox) Suppose we wanted to walk down the Infinite Corridor. To do this, we have to first walk down the first half of the path, but before that, we need to first walk the first quarter of the path, and before that, the first eighth, and so on. There’s always an “earlier” action to take, and “it’s impossible to do infinitely many actions” (says Zeno).

The usual explanation is that we can add up 1 1 1 + + + ··· = 1. 2 4 8 It’s a bit hard to convince the Greeks why this resolves the paradox, though...

2 Some history

Ancient Greeks complained about a lot of mathematics:

Example (Zeno’s paradox) Suppose we wanted to walk down the Infinite Corridor. To do this, we have to first walk down the first half of the path, but before that, we need to first walk the first quarter of the path, and before that, the first eighth, and so on. There’s always an “earlier” action to take, and “it’s impossible to do infinitely many actions” (says Zeno).

The usual explanation is that we can add up 1 1 1 + + + ··· = 1. 2 4 8 It’s a bit hard to convince the Greeks why this resolves the paradox, though...

2 Some history, part 2

In the 19th century, people took a closer look at the foundations of calculus.

• Techniques were useful, but it was hard to resolve how geometric they seemed compared to the algebraic ideas in number theory (like we discussed on days 2-3). • Tangent lines? Drawing rectangles for area? • It also wasn’t clear how to actually use an object like dx, and worse, it wasn’t even clear what the real numbers really were at all.

(Important names include Weierstrass, Cantor, and Dedekind.)

3 Some history, part 2

In the 19th century, people took a closer look at the foundations of calculus.

• Techniques were useful, but it was hard to resolve how geometric they seemed compared to the algebraic ideas in number theory (like we discussed on days 2-3). • Tangent lines? Drawing rectangles for area? • It also wasn’t clear how to actually use an object like dx, and worse, it wasn’t even clear what the real numbers really were at all.

(Important names include Weierstrass, Cantor, and Dedekind.)

3 Some history, part 2

In the 19th century, people took a closer look at the foundations of calculus.

• Techniques were useful, but it was hard to resolve how geometric they seemed compared to the algebraic ideas in number theory (like we discussed on days 2-3). • Tangent lines? Drawing rectangles for area? • It also wasn’t clear how to actually use an object like dx, and worse, it wasn’t even clear what the real numbers really were at all.

(Important names include Weierstrass, Cantor, and Dedekind.)

3 Some history, part 2

In the 19th century, people took a closer look at the foundations of calculus.

• Techniques were useful, but it was hard to resolve how geometric they seemed compared to the algebraic ideas in number theory (like we discussed on days 2-3). • Tangent lines? Drawing rectangles for area? • It also wasn’t clear how to actually use an object like dx, and worse, it wasn’t even clear what the real numbers really were at all.

(Important names include Weierstrass, Cantor, and Dedekind.)

3 The analogy to yesterday’s dumpling analogy

Problem Suppose that I need to create a 1-meter-long meter stick. What does it mean (in the real world) for something to “be 1 meter long?”

Answer: We can try to do physics experiments or compare this stick to other meter sticks, but there’s always some level of desired precision. (Normally there’s some margin of error.)

4 The analogy to yesterday’s dumpling analogy

Problem Suppose that I need to create a 1-meter-long meter stick. What does it mean (in the real world) for something to “be 1 meter long?”

Answer: We can try to do physics experiments or compare this stick to other meter sticks, but there’s always some level of desired precision. (Normally there’s some margin of error.)

4 Key concept of today

We’ll try to develop language that makes talking about limits and infinite sums possible.

Proposition (A general principle) Analysis (limits, sequences, and calculus) centers around being close enough to the final answer we’re aiming for.

(Imagine that an evil construction worker is trying to quality-test our meter sticks – we need to always meet their demands.)

5 Key concept of today

We’ll try to develop language that makes talking about limits and infinite sums possible.

Proposition (A general principle) Analysis (limits, sequences, and calculus) centers around being close enough to the final answer we’re aiming for.

(Imagine that an evil construction worker is trying to quality-test our meter sticks – we need to always meet their demands.)

5 Limits and convergence, part 1

Last lecture, I mentioned that we often want to say things like 1 limn→∞ n = 0 in algebra and calculus. So looking at 1 1 1 1 1 1, , , , , , ··· , 2 3 4 5 6 eventually the limit of this sequence is 0.

Even though none of those numbers are exactly 0, we can still get arbitrarily close to zero if we keep going down our list of numbers.

6 Limits and convergence, part 1

Last lecture, I mentioned that we often want to say things like 1 limn→∞ n = 0 in algebra and calculus. So looking at 1 1 1 1 1 1, , , , , , ··· , 2 3 4 5 6 eventually the limit of this sequence is 0.

Even though none of those numbers are exactly 0, we can still get arbitrarily close to zero if we keep going down our list of numbers.

6 Let’s be a bit more precise

For example, suppose that we want to be below the blue line y = 0.18.

At first, our sequence isn’t good enough, but as long as we’re on the 6th number or later, we’re below the blue line. And even if the blue line gets closer, our sequence will still eventually be within the margin of error.

7 Let’s be a bit more precise

For example, suppose that we want to be below the blue line y = 0.18.

At first, our sequence isn’t good enough, but as long as we’re on the 6th number or later, we’re below the blue line. And even if the blue line gets closer, our sequence will still eventually be within the margin of error.

7 Let’s be a bit more precise

For example, suppose that we want to be below the blue line y = 0.18.

At first, our sequence isn’t good enough, but as long as we’re on the 6th number or later, we’re below the blue line. And even if the blue line gets closer, our sequence will still eventually be within the margin of error.

7 Another example

Suppose that I’m making a bunch of meter sticks, and I’m getting better and better. (If you’re curious, the formula is that the nth number is sin n an = 1 + n .)

None of my meter sticks will be exactly 1 meter long, but here the construction worker wants a margin of error of 0.1, and that’s true starting from the 9th stick.

8 Another example

Suppose that I’m making a bunch of meter sticks, and I’m getting better and better. (If you’re curious, the formula is that the nth number is sin n an = 1 + n .)

None of my meter sticks will be exactly 1 meter long, but here the construction worker wants a margin of error of 0.1, and that’s true starting from the 9th stick.

8 Another example

Suppose that I’m making a bunch of meter sticks, and I’m getting better and better. (If you’re curious, the formula is that the nth number is sin n an = 1 + n .)

None of my meter sticks will be exactly 1 meter long, but here the construction worker wants a margin of error of 0.1, and that’s true starting from the 9th stick.

8 A formal definition

Definition

Suppose we have an infinite sequence of numbers {a1, a2, a3, ···}. This sequence converges to a number a (or has limit a) if for every number ε > 0, the sequence is eventually within ε of a. In other words, we can

find some integer N so that |an − a| <ε for all n ≥ N.

Think of ε as the “evil construction worker’s demand.” Note: because ε is chosen first, N can depend on ε.

9 A formal definition

Definition

Suppose we have an infinite sequence of numbers {a1, a2, a3, ···}. This sequence converges to a number a (or has limit a) if for every number ε > 0, the sequence is eventually within ε of a. In other words, we can

find some integer N so that |an − a| <ε for all n ≥ N.

Think of ε as the “evil construction worker’s demand.” Note: because ε is chosen first, N can depend on ε.

9 A formal definition

Definition

Suppose we have an infinite sequence of numbers {a1, a2, a3, ···}. This sequence converges to a number a (or has limit a) if for every number ε > 0, the sequence is eventually within ε of a. In other words, we can

find some integer N so that |an − a| <ε for all n ≥ N.

Think of ε as the “evil construction worker’s demand.” Note: because ε is chosen first, N can depend on ε.

9 A formal definition

Definition

Suppose we have an infinite sequence of numbers {a1, a2, a3, ···}. This sequence converges to a number a (or has limit a) if for every number ε > 0, the sequence is eventually within ε of a. In other words, we can

find some integer N so that |an − a| <ε for all n ≥ N.

Think of ε as the “evil construction worker’s demand.” Note: because ε is chosen first, N can depend on ε.

9 A formal definition

Definition

Suppose we have an infinite sequence of numbers {a1, a2, a3, ···}. This sequence converges to a number a (or has limit a) if for every number ε > 0, the sequence is eventually within ε of a. In other words, we can

find some integer N so that |an − a| <ε for all n ≥ N.

Think of ε as the “evil construction worker’s demand.” Note: because ε is chosen first, N can depend on ε.

9 A formal definition

Definition

Suppose we have an infinite sequence of numbers {a1, a2, a3, ···}. This sequence converges to a number a (or has limit a) if for every number ε > 0, the sequence is eventually within ε of a. In other words, we can

find some integer N so that |an − a| <ε for all n ≥ N.

Think of ε as the “evil construction worker’s demand.” Note: because ε is chosen first, N can depend on ε.

9 The structure of the proof (with numbers)

Problem 1 Show that the sequence an = n converges to0.

• Suppose we are given (for example) ε = 0.01. We need to find a way to make sure all of the terms an are eventually less than0 .01 1 away from0: this happens whenever n < 0.01 ⇐⇒ n > 100. So let’s pick N = 101 (larger cutoffs work too). 1 • Now we can show that n is eventually close enough to 0. Indeed, 1 1 ≤ < 0.01 n 101 whenever n ≥ 101, which implies that our sequence is eventually always within 0.01 of the point 0.

10 The structure of the proof (with numbers)

Problem 1 Show that the sequence an = n converges to0.

• Suppose we are given (for example) ε = 0.01. We need to find a way to make sure all of the terms an are eventually less than0 .01 1 away from0 : this happens whenever n < 0.01 ⇐⇒ n > 100. So let’s pick N = 101 (larger cutoffs work too). 1 • Now we can show that n is eventually close enough to 0. Indeed, 1 1 ≤ < 0.01 n 101 whenever n ≥ 101, which implies that our sequence is eventually always within 0.01 of the point 0.

10 The structure of the proof (with numbers)

Problem 1 Show that the sequence an = n converges to0.

• Suppose we are given (for example) ε = 0.01. We need to find a way to make sure all of the terms an are eventually less than0 .01 1 away from0 : this happens whenever n < 0.01 ⇐⇒ n > 100. So let’s pick N = 101 (larger cutoffs work too). 1 • Now we can show that n is eventually close enough to 0. Indeed, 1 1 ≤ < 0.01 n 101 whenever n ≥ 101, which implies that our sequence is eventually always within 0.01 of the point 0.

10 The structure of the proof (with numbers)

Problem 1 Show that the sequence an = n converges to0.

• Suppose we are given (for example) ε = 0.01. We need to find a way to make sure all of the terms an are eventually less than0 .01 1 away from0 : this happens whenever n < 0.01 ⇐⇒ n > 100. So let’s pick N = 101 (larger cutoffs work too). 1 • Now we can show that n is eventually close enough to 0. Indeed, 1 1 ≤ < 0.01 n 101 whenever n ≥ 101, which implies that our sequence is eventually always within 0.01 of the point 0.

10 The structure of the proof (with numbers)

Problem 1 Show that the sequence an = n converges to0.

• Suppose we are given (for example) ε = 0.01. We need to find a way to make sure all of the terms an are eventually less than0 .01 1 away from0 : this happens whenever n < 0.01 ⇐⇒ n > 100. So let’s pick N = 101 (larger cutoffs work too). 1 • Now we can show that n is eventually close enough to 0. Indeed, 1 1 ≤ < 0.01 n 101 whenever n ≥ 101, which implies that our sequence is eventually always within 0.01 of the point 0.

10 The structure of the proof (with numbers)

Problem 1 Show that the sequence an = n converges to0.

• Suppose we are given (for example) ε = 0.01. We need to find a way to make sure all of the terms an are eventually less than0 .01 1 away from0 : this happens whenever n < 0.01 ⇐⇒ n > 100. So let’s pick N = 101 (larger cutoffs work too). 1 • Now we can show that n is eventually close enough to 0. Indeed, 1 1 ≤ < 0.01 n 101 whenever n ≥ 101, which implies that our sequence is eventually always within 0.01 of the point 0.

10 The same proof, written out formally

Problem 1 Show that the sequence an = n converges to0.

Proof. • Suppose we are given some arbitrary ε. We need to find a way to make sure all of the terms an are eventually less than ε away from0: 1 1 this happens whenever n <ε ⇐⇒ n > ε . So let’s pick 1 N = 1 + d ε e (larger cutoffs work too). 1 • Now we can show that n is eventually close enough to 0. Indeed, 1 1 ≤ <ε n N whenever n ≥ N, which implies that our sequence is eventually 1 always within ε of the point0 (formally, n − 0 <ε ). Since ε was arbitrary, this completes the proof.

11 The flip side of the coin

Problem n Let’s show the sequence given by an = (−1) does not converge to 1 (in fact, it doesn’t converge at all).

The first few terms of this sequence are

−1, 1, −1, 1, −1, 1, ··· .

(Notice: now we want to prove that convergence fails. So we get to be the evil wizard – what does that mean?)

12 The flip side of the coin

Problem n Let’s show the sequence given by an = (−1) does not converge to 1 (in fact, it doesn’t converge at all).

The first few terms of this sequence are

−1, 1, −1, 1, −1, 1, ··· .

(Notice: now we want to prove that convergence fails. So we get to be the evil wizard – what does that mean?)

12 The flip side of the coin

Problem n Let’s show the sequence given by an = (−1) does not converge to 1 (in fact, it doesn’t converge at all).

The first few terms of this sequence are

−1, 1, −1, 1, −1, 1, ··· .

(Notice: now we want to prove that convergence fails. So we get to be the evil wizard – what does that mean?)

12 The flip side of the coin

Problem n Let’s show the sequence given by an = (−1) does not converge to 1 (in fact, it doesn’t converge at all).

The first few terms of this sequence are

−1, 1, −1, 1, −1, 1, ··· .

(Notice: now we want to prove that convergence fails. So we get to be the evil wizard – what does that mean?)

12 A proof that convergence fails

Problem n Show that the sequence an = (−1) doesn’t converge to 1.

• Suppose for the sake of contradiction that an converges to1. • Now we are the evil wizard: we show that this fails by trying ε = 0.1. (Other choices would work here.) • Now it’s impossible to eventually always be within0 .1 of1 in this sequence (because of the −1s). In other words, it is impossible to choose N so that after the Nth term, we’re always between the blue lines. So convergence fails, as desired. To finish, replace1 with an arbitrary number x (exercise)! 13 A proof that convergence fails

Problem n Show that the sequence an = (−1) doesn’t converge to 1.

• Suppose for the sake of contradiction that an converges to1. • Now we are the evil wizard: we show that this fails by trying ε = 0.1. (Other choices would work here.) • Now it’s impossible to eventually always be within0 .1 of1 in this sequence (because of the −1s). In other words, it is impossible to choose N so that after the Nth term, we’re always between the blue lines. So convergence fails, as desired. To finish, replace1 with an arbitrary number x (exercise)! 13 A proof that convergence fails

Problem n Show that the sequence an = (−1) doesn’t converge to 1.

• Suppose for the sake of contradiction that an converges to1. • Now we are the evil wizard: we show that this fails by trying ε = 0.1. (Other choices would work here.) • Now it’s impossible to eventually always be within0 .1 of1 in this sequence (because of the −1s). In other words, it is impossible to choose N so that after the Nth term, we’re always between the blue lines. So convergence fails, as desired. To finish, replace1 with an arbitrary number x (exercise)! 13 A proof that convergence fails

Problem n Show that the sequence an = (−1) doesn’t converge to 1.

• Suppose for the sake of contradiction that an converges to1. • Now we are the evil wizard: we show that this fails by trying ε = 0.1. (Other choices would work here.) • Now it’s impossible to eventually always be within0 .1 of1 in this sequence (because of the −1s). In other words, it is impossible to choose N so that after the Nth term, we’re always between the blue lines. So convergence fails, as desired. To finish, replace1 with an arbitrary number x (exercise)! 13 A proof that convergence fails

Problem n Show that the sequence an = (−1) doesn’t converge to 1.

• Suppose for the sake of contradiction that an converges to1. • Now we are the evil wizard: we show that this fails by trying ε = 0.1. (Other choices would work here.) • Now it’s impossible to eventually always be within0 .1 of1 in this sequence (because of the −1s). In other words, it is impossible to choose N so that after the Nth term, we’re always between the blue lines. So convergence fails, as desired. To finish, replace1 with an arbitrary number x (exercise)! 13 We always deal with infinity separately

What does it mean to “go to infinity?” (Think about last lecture.)

Definition

A sequence {an} diverges to +∞ if for every real number x, the sequence is eventually larger than x: we can pick an N (in terms of x) so

that an > x for all n ≥ N.

2 Exercise: prove that limn→∞ n = ∞. (Remember: to do this, prove that n2 > x eventually, no matter what value of x we are given.)

14 We always deal with infinity separately

What does it mean to “go to infinity?” (Think about last lecture.)

Definition

A sequence {an} diverges to +∞ if for every real number x, the sequence is eventually larger than x: we can pick an N (in terms of x) so

that an > x for all n ≥ N.

2 Exercise: prove that limn→∞ n = ∞. (Remember: to do this, prove that n2 > x eventually, no matter what value of x we are given.)

14 We always deal with infinity separately

What does it mean to “go to infinity?” (Think about last lecture.)

Definition

A sequence {an} diverges to +∞ if for every real number x, the sequence is eventually larger than x: we can pick an N (in terms of x) so

that an > x for all n ≥ N.

2 Exercise: prove that limn→∞ n = ∞. (Remember: to do this, prove that n2 > x eventually, no matter what value of x we are given.)

14 We always deal with infinity separately

What does it mean to “go to infinity?” (Think about last lecture.)

Definition

A sequence {an} diverges to +∞ if for every real number x, the sequence is eventually larger than x: we can pick an N (in terms of x) so

that an > x for all n ≥ N.

2 Exercise: prove that limn→∞ n = ∞. (Remember: to do this, prove that n2 > x eventually, no matter what value of x we are given.)

14 Sequences to functions; integers to real numbers

Now we’re almost ready to talk about real-valued functions, but first: Fact The issue with rational numbers is that a sequence of rational numbers can converge to something that’s not rational. For example, the sequence 2 , 3 , 5 , 8 , ··· (ratios of consecutive Fibonacci numbers) converges to the 1 2 3 5 √ 1+ 5 golden ratio φ = 2 .

Fact Real numbers are basically all of the possible convergence points for rational sequences – in fact, this is a way to formally define the real numbers!

15 Sequences to functions; integers to real numbers

Now we’re almost ready to talk about real-valued functions, but first: Fact The issue with rational numbers is that a sequence of rational numbers can converge to something that’s not rational. For example, the sequence 2 , 3 , 5 , 8 , ··· (ratios of consecutive Fibonacci numbers) converges to the 1 2 3 5 √ 1+ 5 golden ratio φ = 2 .

Fact Real numbers are basically all of the possible convergence points for rational sequences – in fact, this is a way to formally define the real numbers!

15 Sequences to functions; integers to real numbers

Now we’re almost ready to talk about real-valued functions, but first: Fact The issue with rational numbers is that a sequence of rational numbers can converge to something that’s not rational. For example, the sequence 2 , 3 , 5 , 8 , ··· (ratios of consecutive Fibonacci numbers) converges to the 1 2 3 5 √ 1+ 5 golden ratio φ = 2 .

Fact Real numbers are basically all of the possible convergence points for rational sequences – in fact, this is a way to formally define the real numbers!

15 Sequences to functions; integers to real numbers

Now we’re almost ready to talk about real-valued functions, but first: Fact The issue with rational numbers is that a sequence of rational numbers can converge to something that’s not rational. For example, the sequence 2 , 3 , 5 , 8 , ··· (ratios of consecutive Fibonacci numbers) converges to the 1 2 3 5 √ 1+ 5 golden ratio φ = 2 .

Fact Real numbers are basically all of the possible convergence points for rational sequences – in fact, this is a way to formally define the real numbers!

15 Limits and continuity: a similar recipe

Here are two ways of describing continuity:

• We can draw the graph without picking up our pencil. • Values of outputs f (x) are close as long as our inputs x are close.

This time, instead of eventually being close enough because we go far enough in the sequence, we make sure that we’re close enough once our inputs are close enough.

16 Limits and continuity: a similar recipe

Here are two ways of describing continuity:

• We can draw the graph without picking up our pencil. • Values of outputs f (x) are close as long as our inputs x are close.

This time, instead of eventually being close enough because we go far enough in the sequence, we make sure that we’re close enough once our inputs are close enough.

16 Limits and continuity: a similar recipe

Here are two ways of describing continuity:

• We can draw the graph without picking up our pencil. • Values of outputs f (x) are close as long as our inputs x are close.

This time, instead of eventually being close enough because we go far enough in the sequence, we make sure that we’re close enough once our inputs are close enough.

16 Limits and continuity: a similar recipe

Here are two ways of describing continuity:

• We can draw the graph without picking up our pencil. • Values of outputs f (x) are close as long as our inputs x are close.

This time, instead of eventually being close enough because we go far enough in the sequence, we make sure that we’re close enough once our inputs are close enough.

16 Let’s see what the definition looks like

Definition (The “epsilon-delta”) Suppose we have a function f : R → R. This function has limit L at a point x = c (denoted limx→c f (x) = L) if for every threshold ε> 0, we can pick a δ> 0 (in terms of ε) so that whenever 0 < |x − c| <δ , we

have |f (x) − L| <ε . And f is continuous at x = c if limx→c f (x) = f (c).

17 Let’s see what the definition looks like

Definition (The “epsilon-delta”) Suppose we have a function f : R → R. This function has limit L at a point x = c (denoted limx→c f (x) = L) if for every threshold ε> 0, we can pick a δ> 0 (in terms of ε) so that whenever 0 < |x − c| <δ , we

have |f (x) − L| <ε . And f is continuous at x = c if limx→c f (x) = f (c).

17 Let’s see what the definition looks like

Definition (The “epsilon-delta”) Suppose we have a function f : R → R. This function has limit L at a point x = c (denoted limx→c f (x) = L) if for every threshold ε> 0, we can pick a δ> 0 (in terms of ε) so that whenever 0 < |x − c| <δ , we

have |f (x) − L| <ε . And f is continuous at x = c if limx→c f (x) = f (c).

17 Let’s see what the definition looks like

Definition (The “epsilon-delta”) Suppose we have a function f : R → R. This function has limit L at a point x = c (denoted limx→c f (x) = L) if for every threshold ε> 0, we can pick a δ> 0 (in terms of ε) so that whenever 0 < |x − c| <δ , we

have |f (x) − L| <ε . And f is continuous at x = c if limx→c f (x) = f (c).

17 Connecting the two definitions

As we said earlier,all real numbers can be worked with (or defined as) limits of rational sequences.

This is called sequential continuity: in both cases, we care about eventually being within ε of the final goal. (This is one reason we use the notation “x → c.”) And tying things back to day 6, I like to think of testing set sizes with functions as similar to testing continuity with sequences. 18 Connecting the two definitions

As we said earlier,all real numbers can be worked with (or defined as) limits of rational sequences.

This is called sequential continuity: in both cases, we care about eventually being within ε of the final goal. (This is one reason we use the notation “x → c.”) And tying things back to day 6, I like to think of testing set sizes with functions as similar to testing continuity with sequences. 18 Connecting the two definitions

As we said earlier,all real numbers can be worked with (or defined as) limits of rational sequences.

This is called sequential continuity: in both cases, we care about eventually being within ε of the final goal. (This is one reason we use the notation “x → c.”) And tying things back to day 6, I like to think of testing set sizes with functions as similar to testing continuity with sequences. 18 Derivatives really bring this into perspective

Think about what we’re saying when we write something like

f (x + h) − f (x) f 0(x) = lim . h→0 h

• Saying h → 0 means that the limit exists no matter how we approach 0 (in other words, no matter which sequence {h1, h2, ···} converging to 0 we use). • We’ve removed the need for infinitesimals: we’re not actually evaluating the function f (x) at a point x + dx. • The fact that this derivative can exist makes many theorems – the power rule, chain rule, u-substitution, fundamental theorem of calculus – actually provable! (Try proving these now.)

19 Derivatives really bring this into perspective

Think about what we’re saying when we write something like

f (x + h) − f (x) f 0(x) = lim . h→0 h

• Saying h → 0 means that the limit exists no matter how we approach 0 (in other words, no matter which sequence {h1, h2, ···} converging to 0 we use). • We’ve removed the need for infinitesimals: we’re not actually evaluating the function f (x) at a point x + dx. • The fact that this derivative can exist makes many theorems – the power rule, chain rule, u-substitution, fundamental theorem of calculus – actually provable! (Try proving these now.)

19 Derivatives really bring this into perspective

Think about what we’re saying when we write something like

f (x + h) − f (x) f 0(x) = lim . h→0 h

• Saying h → 0 means that the limit exists no matter how we approach 0 (in other words, no matter which sequence {h1, h2, ···} converging to 0 we use). • We’ve removed the need for infinitesimals: we’re not actually evaluating the function f (x) at a point x + dx. • The fact that this derivative can exist makes many theorems – the power rule, chain rule, u-substitution, fundamental theorem of calculus – actually provable! (Try proving these now.)

19 Derivatives really bring this into perspective

Think about what we’re saying when we write something like

f (x + h) − f (x) f 0(x) = lim . h→0 h

• Saying h → 0 means that the limit exists no matter how we approach 0 (in other words, no matter which sequence {h1, h2, ···} converging to 0 we use). • We’ve removed the need for infinitesimals: we’re not actually evaluating the function f (x) at a point x + dx. • The fact that this derivative can exist makes many theorems – the power rule, chain rule, u-substitution, fundamental theorem of calculus – actually provable! (Try proving these now.)

19 Derivatives really bring this into perspective

Think about what we’re saying when we write something like

f (x + h) − f (x) f 0(x) = lim . h→0 h

• Saying h → 0 means that the limit exists no matter how we approach 0 (in other words, no matter which sequence {h1, h2, ···} converging to 0 we use). • We’ve removed the need for infinitesimals: we’re not actually evaluating the function f (x) at a point x + dx. • The fact that this derivative can exist makes many theorems – the power rule, chain rule, u-substitution, fundamental theorem of calculus – actually provable! (Try proving these now.)

19 From here, what does analysis look like?

Two main ways to keep going:

• Increase abstraction. Instead of just talking about real numbers, talk about limits, functions, derivatives in n-dimensional space. Or think about more general spaces (metric spaces, topological spaces). • Study more properties. Think more about implications of continuity, classifying functions, understanding what to do with them.

We’ll end this class by showing you some “sneak peeks.”

20 From here, what does analysis look like?

Two main ways to keep going:

• Increase abstraction. Instead of just talking about real numbers, talk about limits, functions, derivatives in n-dimensional space. Or think about more general spaces (metric spaces, topological spaces). • Study more properties. Think more about implications of continuity, classifying functions, understanding what to do with them.

We’ll end this class by showing you some “sneak peeks.”

20 From here, what does analysis look like?

Two main ways to keep going:

• Increase abstraction. Instead of just talking about real numbers, talk about limits, functions, derivatives in n-dimensional space. Or think about more general spaces (metric spaces, topological spaces). • Study more properties. Think more about implications of continuity, classifying functions, understanding what to do with them.

We’ll end this class by showing you some “sneak peeks.”

20 From here, what does analysis look like?

Two main ways to keep going:

• Increase abstraction. Instead of just talking about real numbers, talk about limits, functions, derivatives in n-dimensional space. Or think about more general spaces (metric spaces, topological spaces). • Study more properties. Think more about implications of continuity, classifying functions, understanding what to do with them.

We’ll end this class by showing you some “sneak peeks.”

20 From here, what does analysis look like?

Two main ways to keep going:

• Increase abstraction. Instead of just talking about real numbers, talk about limits, functions, derivatives in n-dimensional space. Or think about more general spaces (metric spaces, topological spaces). • Study more properties. Think more about implications of continuity, classifying functions, understanding what to do with them.

We’ll end this class by showing you some “sneak peeks.”

20 Sneak peek 1: power series aren’t so great

• From 18.01: Taylor series help us write many functions as infinite x3 polynomials (e.g. sin x = x − 6 + ··· ). • Often we treat functions and their Taylor series as the same. • But consider this function: e−1/x for x > 0 and 0 otherwise.

All of its derivatives at x = 0 are zero, so the Taylor series is just 0...

Key concept: There are many different descriptions of “niceness” for functions!

21 Sneak peek 1: power series aren’t so great

• From 18.01: Taylor series help us write many functions as infinite x3 polynomials (e.g. sin x = x − 6 + ··· ). • Often we treat functions and their Taylor series as the same. • But consider this function: e−1/x for x > 0 and 0 otherwise.

All of its derivatives at x = 0 are zero, so the Taylor series is just 0...

Key concept: There are many different descriptions of “niceness” for functions!

21 Sneak peek 1: power series aren’t so great

• From 18.01: Taylor series help us write many functions as infinite x3 polynomials (e.g. sin x = x − 6 + ··· ). • Often we treat functions and their Taylor series as the same. • But consider this function: e−1/x for x > 0 and 0 otherwise.

All of its derivatives at x = 0 are zero, so the Taylor series is just 0...

Key concept: There are many different descriptions of “niceness” for functions!

21 Sneak peek 1: power series aren’t so great

• From 18.01: Taylor series help us write many functions as infinite x3 polynomials (e.g. sin x = x − 6 + ··· ). • Often we treat functions and their Taylor series as the same. • But consider this function: e−1/x for x > 0 and 0 otherwise.

All of its derivatives at x = 0 are zero, so the Taylor series is just 0...

Key concept: There are many different descriptions of “niceness” for functions!

21 Sneak peek 2: continuity and infinity

Interesting idea: what if we try to take limits of functions, instead of just numbers? Consider the functions f (x) = x n for various n (with domain [0, 1]).

As n gets larger and larger, the functions go to 0, except at the point x = 1. So the limit is not continuous! Key concept: Continuity is a subtle topic (more at recitation)!

22 Sneak peek 2: continuity and infinity

Interesting idea: what if we try to take limits of functions, instead of just numbers? Consider the functions f (x) = x n for various n (with domain [0, 1]).

As n gets larger and larger, the functions go to 0, except at the point x = 1. So the limit is not continuous! Key concept: Continuity is a subtle topic (more at recitation)!

22 Sneak peek 2: continuity and infinity

Interesting idea: what if we try to take limits of functions, instead of just numbers? Consider the functions f (x) = x n for various n (with domain [0, 1]).

As n gets larger and larger, the functions go to 0, except at the point x = 1. So the limit is not continuous! Key concept: Continuity is a subtle topic (more at recitation)!

22 Sneak peek 3: what came out of this rigorization?

There’s a reason analysis is still an ongoing field of research.

• Measure theory: how do we measure area and volume formally? How do we prove that “most” real numbers are irrational? • Functional analysis: what happens if we go meta and study functions as points in space? • Hilbert’s program: if we can do calculus rigorously now, can we derive everything in math from the axioms? (This ultimately leads to the work of G¨odel.)

Key concept: new mathematical ideas are invented by coming up with new perspectives.

23 Sneak peek 3: what came out of this rigorization?

There’s a reason analysis is still an ongoing field of research.

• Measure theory: how do we measure area and volume formally? How do we prove that “most” real numbers are irrational? • Functional analysis: what happens if we go meta and study functions as points in space? • Hilbert’s program: if we can do calculus rigorously now, can we derive everything in math from the axioms? (This ultimately leads to the work of G¨odel.)

Key concept: new mathematical ideas are invented by coming up with new perspectives.

23 Sneak peek 3: what came out of this rigorization?

There’s a reason analysis is still an ongoing field of research.

• Measure theory: how do we measure area and volume formally? How do we prove that “most” real numbers are irrational? • Functional analysis: what happens if we go meta and study functions as points in space? • Hilbert’s program: if we can do calculus rigorously now, can we derive everything in math from the axioms? (This ultimately leads to the work of G¨odel.)

Key concept: new mathematical ideas are invented by coming up with new perspectives.

23 Sneak peek 3: what came out of this rigorization?

There’s a reason analysis is still an ongoing field of research.

• Measure theory: how do we measure area and volume formally? How do we prove that “most” real numbers are irrational? • Functional analysis: what happens if we go meta and study functions as points in space? • Hilbert’s program: if we can do calculus rigorously now, can we derive everything in math from the axioms? (This ultimately leads to the work of G¨odel.)

Key concept: new mathematical ideas are invented by coming up with new perspectives.

23 Summary

• Back-and-forth game in the definitions of limits and continuity. • Analysis lets us go back to calculus and understand concepts at a deeper level. • It’s always possible to go farther into the rabbit hole!

24 Thank you for taking our class!

Special thanks to:

• Columbia’s Proof Writing Workshop, • Eric Baer (IAP 2015), • Professor Larry Guth, • Mentors and graders, • and many other people!

See you all one more time at recitation :)

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