existence of the minimizer for a finite dimensional subspace of a normed vector space Let $X$ be a normed vector space and let $Y$ be a finite dimensional subspace. If $x \in Y^C$, then we need to prove that there exist $y_0 \in Y$ such that $||x-y_0||= d(x,Y)= \inf_{y\in Y}||x-y||=M$.
My work: by the definition of the infimum, we have for each $n \in \mathbb{N}$, an element $y_n\in Y$ such that:
$||x-y_n|| < M +\frac{1}{n}$
but, $ M\leq ||x-y_n||$ thus we let $n\rightarrow \infty$ and we get:
$\lim_{n\rightarrow \infty}||x-y_n||=M$.
I need to show that this sequence $\{y_n\}$ has a convergent subsequence by proving that it is bounded. I was able to prove $\{y_n\}$ is bounded, but not sure how to show that it has a convergent subsequence.

In a finite-dimensional normed space, closed bounded balls are compact.
