Distance between a point and closed set in finite dimensional space
Solution 1:
Suppose $X$ infinite dimensional. Then the unit sphere $S$ is not compact (Riesz theorem), and therefore there is a sequence $x = (x_n)_n$ on $S$ without accumulation points. Denote by $x'$ the new sequence defined by $x_n' = (1 + \frac{1}{n})x_n$.
Since $x'$ and $x$ have the same accumulation points, $x'$ doesn't have any. So the set $C$ of the values of $x'$ is closed.
Now $d(0, C) = 1$, and there is no point in $C$ of norm $1$.