closure of ball in metric topology(usual)

Question

Let $(X,d)$ be a metric space with metric usual topology $\tau$ Let $x\in X$ and $r>0$ then prove that $cl(S_r(x))=\{y\in X, d(x,y)\le r\}$ $cl(X)=$closure of X

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$\{y\in X, d(x,y)\le r\}$ is closed set but how can we prove the equality ???//

Answer 1

You cannot prove it, since it is not true (in general). If, say, $X=\{0,1\}$, with its usual distance, then $\overline{S_1(1)}=\{1\}$, but $\{x\in X\mid d(x,1)\leqslant1\}=\{0,1\}$.