Reflexive Banach Space: For any $x ∈ E$, there exists $x_0 ∈ M$ so that $|x−x_0| =\inf\{|x−y| :y ∈ M \}$

Let $E$ be a reflexive $\mathbb K$−Banach space and assume that $M$ is a convex, closed, bounded subset of $E.$ Prove that for any $x ∈ E$, there exists $x_0 ∈ M$ so that $|x−x_0| = \inf\{|x−y| :y ∈ M \}$.

I got the hint to use Fenchel-Rockafellar Theorem (also known as Fenchel's Duality Theorem) and then use the weak* topology $\sigma(E^*,E)$.

But I don't see the use of applying this theorem in this case. Why does one need this and how does one prove this statement using this particular theorem?

• I find it simpler to look at $$\bigcap_{r > \operatorname{dist}(x,M)} (\overline{B_r(x)}\cap M).$$ – Daniel Fischer Nov 17 '16 at 22:49
• @DanielFischer You suggest to prove the Weierstraß theorem instead of refering to it? Because the way it is proved is precisely looking at those sets and then take the cluster point as $x_0$. My point was to leave some details to OP to discover. – A.Γ. Nov 17 '16 at 23:01