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!1 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!1 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 fa1; a2; a3; · · · g. This sequence converges to a number a (or has limit a) if for every number " > 0, the sequence is eventually within " of a. In other words, we can find some integer N so that jan − aj <" for all n ≥ N.