# how to prove that intersection of compact sets is compact

By Cantor's intersection theorem I know that a sequence of nonempty compact sets which are nested has nonempty intersection. But how can I use that to prove that arbitrary intersection of compact sets is compact?

• If you're just talking about compact sets of the complex plane, just use the Heine-Borel theorem. – Vik78 Oct 24 '16 at 22:31

• Supose we have $\{K_i\}_i a colection of compact sets. This means that for every$i$there exists some$R_i>0$such that$K_i$is bounded:$K_i\subseteq B(0,R_i)$. Now let$K$be the intersection of the$K_i$. By property of intersections$K\subseteq K_i$for all$i$, this implies,$K\subseteq B(0, R_i)$for all$i$, i.e.$K\$ is bounded. – N. Bitar Sep 15 '17 at 14:20