Math Challenges

Submissions for Problem #39

Problem #39

Prove that K is compact directly using the definition.

\[ \displaylines{K\subset\mathbb{R}\\ K=\left\lbrace0\right\rbrace\cup\left\lbrace\frac{1}{n}:n=1,2,3,\ldots\right\rbrace} \]

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zapwai

Solution: \( \)

Explanation:
Given any covering {W_i} of K, at least one of the sets contains {0}, call this set V0. Since 0 is a limit point for the sequence {1/n}, V0 also contains all but finitely many elements of that sequence. Let V1, V2, ... Vn be sets in the covering which contain these missing elements. {Vi} is a finite cover of K.