- Thread starter
- #1

- Feb 5, 2012

- 1,621

Here's a question that I couldn't find the full answer. Any ideas will be greatly appreciated.

I felt that the compactness of \(F_1\) and \(F_2\) could be brought into the question using the following equivalency. However all my attempts to solve the question weren't successful.Let \((X,\,d)\) be a metric space. Let \(F_1,\,F_2\) be two nonempty compact subsets of \(X\). Prove that, there exists \(x_1\in F_1,\,x_2\in F_2,\) such that,

\[d(x_1,\,x_2)=\mbox{inf}\{d(x,\,y):x\in F_1,\,y\in F_2\}\]

Let \((X,\,d)\) be a metric space and \(S\) is a compact subspace of \(X\). Then, any sequence of points in \(S\) has a subsequence which is convergent to a point in \(S\).