Real Analysis

Math 310  MWF 10:30-11:25am
Fall 1996  127 Fowler

Ron Buckmire ∼ 402 Fowler ∼ 259-2536 ∼ ron@oxy.edu ∼ Mon 1:30-3:30pm Wed & Thu 1:30-2:30pm

#16 Friday (10/18/96)

SUMMARY Subsequences

READING pages 48-55

\underline{Warm-Up} Given the sequence s_{n} := 3 + 2(−1)^{n} write down a definition for (t_{k}), a subsequence of (s_{n}) which takes on a single value (i.e. is constant).

\underline{Theorem} 11.2 If A sequence (s_{n}) converges, then every subsequence (s_{n_{k}}) converges to the same limit PROOF:

We are going to prove the famous Bolzano-Weierstrass Theorem, which is important later on when we are dealing with continuity of functions. \underline{Bolzano-Weierstrass Theorem}: Every bounded sequence has a convergent subsequence.

First we need to prove the following theorem. \underline{Theorem} 11.3 Every sequence (s_{n}) has a monotonic subsequence. PROOF:

Bolzano-Weierstrass Theorem 11.5 Every bounded sequence has a convergent subsequence. PROOF:

\underline{Corollary} 11.4 Every sequence (s_{n}) has a monotonic subsequence which converges to lim sup s_{n} and another one which converges to lim inf s_{n}

\underline{Definition} A subsequential limit of (s_{n}) is any real number, +∞ or −∞ which is the limit of some subsequence of (s_{n})

\underline{Example} 11.3 a_{n} = cos(nπ/3)

1. Give an example of a monotone subsequence

2. Write down the set of subsequential limits

3. Write down lim inf a_{n} and lim sup a_{n}

4. Does a_{n} converge? diverge to +∞? diverge to −∞? N.O.T.A.?

5. Is a_{n} bounded?

Homework Assignment (Due Monday October 21) §10, page 47-48: 10.1, 10.6 10.10 §11, page 55: 11.2