;;;; 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.]

;;;; END: "../AdvCalc/eprobs.txt" ;;;;