1 answer

Real Analysis II Please do it without using Heine-Borel's theorem and do it only if you're sure Problem: Let E

Question:

Real Analysis II
Please do it without using Heine-Borel's theorem and do it only if you're sure
Problem: Let E be a closed bounded subset of En and r be any function mapping E to
(0,∞). Then there exists finitely many points yi ∈ E, i = 1,...,N such that

E CUBy

Here Br(yi)(yi) is the open ball (neighborhood) of radius r(yi) centered at yi.

Also, following definitions & theorems should help that
Definition. A subset S of a topological space So is compact if every open covering of S contains a finite subcovering If So, Theorem 2.9. A set S E is compact ifand only ifS is closed and bounded. Theorem 2.10. If S is a compact set and f is continuous on S, then f(S) is a compact set.

E CUBy
Definition. A subset S of a topological space So is compact if every open covering of S contains a finite subcovering If So, considered as a subset of itself, is compact, then So is a compact topological space.
Theorem 2.9. A set S E is compact ifand only ifS is closed and bounded.
Theorem 2.10. If S is a compact set and f is continuous on S, then f(S) is a compact set.

Answers

E ヒ be closed boundet Subset(f n :E→R be ts closed A an boundad TCy)

.

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