Question

By using the definition of compact set in R, show that a closed subset X of a compact set K is compact.

Answer 1

http://en.wikibooks.org/wiki/Topology/Compactness#Important_Properties

Answer 2

\[\forall \left\{ x_n \right\} \subset X\], then \[\left\{ x_n \right\} \subset K\] and because K is a compact set, there exists a subsequence \[\left\{ x_{n_k} \right\}\] converging to \[x \in K\]. Moreover, X is a closed set, so \[x \in X\].
Hence X is a compact set.

Answer 3

Big thanks to both of you @estudier and @anhkhoavo1210 :D