Problem:

Let K be a sequentially compact subset of and suppose that O(this is the letter 'O') is an open subset of that contains K. Prove that there exists some positive number r such that for any point ,

Proof:

(I have a hard time with these "prove that there exists..." type proofs)

Let O be is an open subset of that contains K, which is sequentially compact.

Pick a sequence such that and such that since

Since K is sequentially compact, then there is a subsequence .

Let

then