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 y_i ∈ E, i = 1,...,N such that

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

Also, following definitions & theorems should help that

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.

Answer 1

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