Theorem
Sequentially compact subsets of a metric space are totally bounded.
Proof
Let
be a subset of a metric space
Supposeis not totally bounded, then
exists such that no
- net exists. Let
exists such that
otherwise
would be- net.
exists such that
otherwise
would be- net.
Continue this procedure to obtain a set of points
such that
for
This sequence does not have a convergent subsequence, henceis not sequentially compact.