Show that "the distance from $p_0$ to $S$" exists

Question

Prove that if $S$ is non-empty compact subset of metric space $E$ and $p_0\in E$, then $\min \{d(p_0,p) : p\in S\}$ exists.

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Since $S$ is compact, it will be enough to prove that function $S\ni p\rightarrow d(p_0,p)\in \mathbb{R}$ is continuous so it attains minimum.

For any $x$ in $S$ and any $\epsilon>0$ choose $\delta=\epsilon$. For any $y\in S$ if we have $|d(p_0,x)-d(p_0,y)|\le d(x,y)<\delta=\epsilon$ then $|d(p_0,x)-d(p_0,y)|<\epsilon$. Is this enough?

Answer 1

Yes, this even shows that the function $d(x, p_0)$ defined on $E$ (or also $S$) is uniformly continuous and so you can apply the theorem on attained minima.