MAT 37518.3Three Important Theorems

MAT 3751 8.3 Three Important Theorems Monotone Sequences

Let {an} .

1. {an} is increasing if n< m蓿 an a m for all n, m N .

2. {an} is strictly increasing if n< m� an a m for all n, m N .

3. {an} is decreasing if n< m蕹 an a m for all n, m N .

4. {an} is strictly decreasing if n< m� an a m for all n, m N .

5. {an} is monotone if it is either increasing or decreasing.

6. {an} is strictly monotone if it is either strictly increasing or strictly decreasing.

Different?

{an} Vs {an | n N}

Theorem 8.3.3

Let {an} .

1. If {an} is increasing and bounded above, then {an} is convergent and

liman= sup{ a n | n } n N .

a lim an = 2. If { n} is increasing and not bounded above, then n .

3. If {an} is decreasing and bounded below, then {an} is convergent and

liman= inf{ a n | n } n N .

a lim an = - 4. If { n} is decreasing and not bounded below, then n .

Corollary 8.3.4 (Monotone Convergence Theorem)

Let {an} . Suppose that {an} is monotone. Then {an} is convergent if and only if

{an} is bounded.

Last Edit: 6/27/2018 5:34 PM Subsequences

Let {an} . Suppose that {an} is defined by f : N such that f( n) = an for all n N . Let g : N N . Suppose that g is strictly increasing. Then the sequence defined by f° g : N is a subsequence of {an} . The sequence defined by f° g is written as an , where g( k) = nk for all k . { k }k =1 N

Example 1 a 1 n { n} ={( - ) }

Example 2

{an} = { n}

Lemma 8.3.7 a a a a a Let { n} , and let { nk } be a subsequence of { n} . If { n} is convergent, then { nk } is convergent and

liman= lim a n kk n  .

2 Lemma 8.3.8

Let {an} . Then {an} has a monotone subsequence.

Theorem 8.3.9 (Bolzano–Weierstrass Theorem)

Let {an} . If {an} is bounded, then {an} has a convergent subsequence.

Cauchy Sequence

Let {an} . {an} is a Cauchy sequence if for each e > 0 , there is some N N such that n, m N imply an- a m < e .

Theorem 8.3.12

Let {an} . If {an} is convergent, then {an} is a Cauchy sequence.

Lemma 8.3.13

Let {an} . Suppose that {an} is a Cauchy sequence. If {an} has a convergent

a a liman= lim a n subsequence { nk }, then { n} is convergent and n k  k .

Lemma 8.3.14

Let {an} . If {an} is a Cauchy sequence, then {an} is bounded.

Theorem 8.3.15

Let {an} . If {an} is a Cauchy sequence, then {an} is convergent.

Corollary 8.3.16 (Cauchy Completeness Theorem)

Let {an} . {an} is convergent if and only if {an} is a Cauchy sequence.

3 Least Upper Bound Property, The Completeness Axiom Let A . If A is nonempty and bounded above, then A has a least upper bound.

Theorem 8.3.17 The following are equivalent. 1. The Least Upper Bound Property. 2. The Monotone Convergence Theorem. 3. The Bolzano–Weierstrass Theorem.