Problem:

Let K be a sequentially compact subset of $\displaystyle \mathbb{R}^n$ and suppose that O(this is the letter 'O') is an open subset of $\displaystyle \mathbb{R}^n$ that contains K. Prove that there exists some positive number r such that for any point $\displaystyle u \in K $ , $\displaystyle B_{r} (u) \subseteq O $

Proof:

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

Let O be is an open subset of $\displaystyle \mathbb{R}^n$ that contains K, which is sequentially compact.

Pick a sequence $\displaystyle x_{n} \in \mathbb{R}^n $ such that $\displaystyle \forall n \in N $ and $\displaystyle u \in K $ such that $\displaystyle u \in B_{\frac{1}{n}}(x_{n}) $ since $\displaystyle \left( \frac{1}{n} > 0 \right)$

Since K is sequentially compact, then there is a subsequence $\displaystyle x_{n_k} \mapsto u $.

Let $\displaystyle \frac{1}{n} < \frac{r}{2}$

then

$\displaystyle B_{\frac{1}{n}} \left( x_{n_k} \right) \subseteq B_{r}(u) \subseteq O $