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> Is it true that if $Y \subset X$ is a closed set and $x \notin Y$ then $\exists$ an element $y\_0 \in Y$ $s.t.$ $d(x,Y)=d(x,y\_0)$ where $d(x,Y)= inf \{d(x,y) : y \in Y \}$ and $d: X \times X \to \mathbb R$ is a metric.
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I was looking the proof of Riesz lemma and something like it came across in it. If the above is true then I will get some result in the proof.

Please someone help..

Thank you..