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

#17 Monday (10/21/96)

SUMMARY Subsequences and lim inf and lim sup READING pages 54-60 \underline{Warm-Up} (5-10 minutes) For each of the sequences find the set of subsequential limits.

(a) Let (r_{n}) be a list of all the rational numbers.

(b) Let (b_{n}) = n[1 + (−1)^{n}] for n ∈ ℕ

\underline{Theorem} 11.7 Let (s_{n}) be any sequence in ℝ and S denote the set of subsequential limits of (s_{n})

(i) S is nonempty (ii) sup S = lim sup s_{n} and inf S = lim inf s_{n} (iii) lim s_{n} exists if and only if S has exactly one element, namely lim s_{n} DISCUSS why each of these three statements “makes sense” to you...

\underline{Theorem} 11.8 Let S denote the set of subsequential limits of a sequence (s_{n}). Suppose (t_{n}) is a sequence in S ∩ ℝ and that t = lim t_{n}. Then t belong to S. Explain this theorem in your own words. What does it mean?

Look at Problem 11.10 on page 56 for an example of where this theorem is useful.

\underline{Definition} A set like S (one which contains all of its limits) is called a CLOSED set. lim inf and lim sup Recall the definition of lim inf s_{n} and lim sup s_{n} of a sequence (s_{n}) (visualize the picture!)

\underline{Theorem} 12.2 Let (s_{n}) be any sequence of any nonzero reals. Then:

lim inf |\frac{s_{n+1}}{s_{n}} | ≤ lim inf |s_{n}|^{1/n} ≤ lim sup |s_{n}|^{1/n} ≤ lim sup |\frac{s_{n+1}}{s_{n}} |

\underline{Theorem} 12.3 If lim |\frac{s_{n+1}}{s_{n}} | exists and equals L, then lim |s_{n}|^{1/n} exists and equals L.

Next: Chapter 14 and Series! Homework Assignment (Due Friday October 25) §11, page 56: 11.4, 11.5 §12, page 60: 12.3, 12.14(a)