Given a point $x$ and a closed subspace $Y$ of a normed space, must the distance from $x$ to $Y$ be achieved by some $y\in Y$? I think no. And I am looking for examples. I would like a sequence $y_n$ in $Y$ such that $||y_n-x||\rightarrow d(x,Y)$ while $y_n$ do not converge. 
Can anyone give a proof or an counterexample to this question?

There cannot be an example in $c_0 \subset \ell^\infty$. In fact $c_0$ is a closed ideal in the $C^*$-algebra $\ell^\infty$. For such pairs distance to ideal always attained be an element in the ideal.
