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For a point $x$ and a non-empty subset $A$ of a metric space $(X, d)$, define $\begin{align}\inf\left\{ d(x,a):a\in A\right\}\end{align}$

Prove that if $y$ is another point in $X$ then $$d(x,A)\leqslant d(x,y)+d(y,A)$$