# University of California Mathematics Questionnaire

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Let
X:=
=
{(an)n21 CR; ƏN E N such that yn>N:an = 0},
and let doc((an)n21, (bn)n21) := supn=1 lan – bnl.
– .
in
(a) (3 points) Prove that (X, doo) is a metric space.
(b) (4 points) Is the metric space (X, doo) complete? (Prove your answer).
Problem 2. 7pts.
Let
=
=
X := {(an)n>1 CR; ƏN E N such that V n >N: An = 0},
and let doo(lan)n21, (bn)n>1) := supn>1 an – Enl.
.
(a) (3 points) Prove that (X, doo) is a metric space.
(b) (4 points) Is the metric space (X, doo) complete? (Prove your answer).
Problem 3. 7pts.
i.) (4 points) Let (X, d) be a metric space, and let K1, K2, …, Kn be
a finite family of compact subsets of X. Proved that 0–1 Ki is a
compact set.
ii.) (3 points) Let (X, d) be a compact metric space. Prove or give a
counterexample: If Ki C K2 C K3 C… is an increasing sequence
of compact subsets of X. Then Um=1Kn is compact.
n
Problem 4. 9pts.
i.) (4 points) Let (X, d) be a metric space. Suppose (2n)nen is a sequence
in X which converges to x E X. Let S := {Xn; n E N} U {x}. Show
that S is compact. (Note: You can use any of the characterizations
of compactness from class).
ii.) (5 points) Let (X, d) be a metric space, let K CX be compact and
let C C X be closed. Assume that
inf{d(x, y); x E K,Y EC} = 0.
х
Prove that KNC +0.

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4 Questions

Tags:
mathematics

metric space

sequence

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