In a compact metric space, a sequence with a certain property is convergent

The question:

Let $(K,d)$ be a compact metric space, if ${x_n}$ is a sequence such that ${d(x_n,x)}$ converges for all $x in K$, then ${x_n}$ is a convergent sequence.

My attempt:

Notice that we can define for each $n$, $f_n(x)=d(x_n,x)$, this is an isometry and thus these are equicontinuous functions.

Also, each converging sequence gives us the fact that these functions are pointwise bounded.

Thus we can us Arzela-Ascoli Therom to show that in fact the sequence of functions ${f_n}$ has a unifromly convergent subsequence.

Now if I can show that their limit is indeed of the form $f(x)=d(a,x)$ for some $ain K$, then I can show that $a$ is the limit of the sequence, but I don’t know how to do it or even if it is true.