Metric Space convergence for $D(x,S)=\inf\{d(x,y)|y\in S\}$

Question

Let $(X,d)$ be a metric space with $S$ a non-empty subset of $X$. For $x\in X$ we define the distance $D$ between $x$ and $S$ as $D(x,S)=\inf\{d(x,y)|y\in S\}$.

I want to show that there are $y_n$ ∈ S so that the sequence of real numbers $r_n=d(x,y_n)$ converges to D.

I have already proved that for S with finite elements. Furthermore, I managed to show that our sequence can be made in a monotonic way such that it is bounded by D at the bottom and on the top by the max $r_n$ (which implies that it converges). However, I do not know how to show its convergence to D.

Thank you in advance

Answer 1

This has nothing to do with metric spaces. It is just the definition of infimum. If $A$ is a set of numbers in $[0, \infty)$ the there is a sequence $a_n$ in $A$ converging to the infimum.