;;;; STA: "../AdvCalc/eprobs.txt" ;;;;
From lfurman@ufl.edu Mon Sep 27 13:39 EDT 1999
Abbreviations: "seq" sequence. "\infty" infinity.
The "extended reals" means the closed interval [ -\infty, +\infty ].
E6. Fix a sequence b-vec of reals, (b_0, b_1, b_2, ...).
For each N in the natural numbers, let
U_N := sup{b_k | k>=N} and
D_N := inf{b_k | k>=N}.
["U" is for "Up" and "D" is for "Down".] Prove,
for all indices L and M, that U_L >= D_M.
E7. Shadow Lemma: Fix a sequence b-vec in the extended reals. Show
that b-vec has a monotonic subsequence.
E8. Give a formal proof, for an arbitrary subset S of extended reals, that
A1: inf{-s | s in S} = -sup{s | s in S} .
E9. Write a formal proof of the Shadow Lemma, along the lines
demonstrated in class.
E10. Suppose that a sequence b-vec in the reals is increasing and
upper-bounded. Suppose that
U := sup(b-vec) is finite.
Prove that lim(b-vec) exists and equals U.
/--------------------------------------------------------\
DEFINITION: Say that (n_j)_{j=1}^\infty is a _list_, if
indices
n_1 < n_2 < n_3 < ... .
We use this definition. when talking about indexing
subsequences of a given sequence.
\________________________________________________________/
E11. Consider a compact set S of extended reals. Suppose that
A2: a-vec, b-vec, ... c-vec
is a list of sequences of points from S, and suppose that our list
has K many sequences.
Prove that there is a list of indices
A3: (n_j)_{j=1}^\infty
such that /each/ sequence of (A2) converges along list (A3). That is
a_{n_1}, a_{n_2}, a_{n_3}, a_{n_4}, ...
converges. So too does
b_{n_1}, b_{n_2}, b_{n_3}, b_{n_4}, ...
and ... and
c_{n_1}, c_{n_2}, c_{n_3}, c_{n_4}, ... .
[Hint: You will want to prove this by induction on K. The key
step is proving it for K=2.]
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