Lebesgue number lemma

Let be a Sequentially Compact Metric Space with open cover .
We can find a s.t. for any there is a s.t. .

Proof

Suppose there is no such .
For any find s.t. there is no s.t. .
As is sequentially compact,
find a convergent subsequence where .
Suppose it converges to .
Then we find an open which contains .
Hence, there is an open ball for .
But for large enough , certainly which is a contradiction.