Show that if \(X\) is a metric space and \(A\subseteq X\) is compact then \(A\) is bounded.
Solution
If \(A=\varnothing\), it is bounded. Assume \(A\ne\varnothing\) and fix \(x_0\in A\).
1. The nested balls\[ B(x_0,1),B(x_0,2),\dots \]form an open cover of \(A\).
2. Compactness gives a finite subcover. If \(N\) is the largest radius occurring, then\[ A\subseteq B(x_0,N). \]
3. Therefore for all \(x,y\in A\),\[ d(x,y)\le d(x,x_0)+d(x_0,y)<2N. \]Thus \(A\) is bounded.