If $d(x_n,A)\to0$ can we conclude $(x_n)$ has a cluster point in $A?$

Question

This question occured to me which I cannot resolve.

Let $(x_n)$ is a sequence in a metric space $(X,d)$ and $A$ be a compact subset of $X.$ If $d(x_n,A)\to0$ can we conclude $(x_n)$ has a cluster point in $A?$

I think it will be true, may be closeness is playing the vital role. But I cannot give a rigourous proof.

Answer 1

If $(x_n)$ is allowed to be in $A$, then not necessarily. For example, we could take a singleton set $A$, and let $(x_n)$ be constant in $A$. Otherwise it's true, and a proof follows.

For each $n$, let $a_n \in A$ such that $d(x_n, a_n) < d(x_n, A) + 2^{-n}$. Then we have a subsequence $(a_{n_k})$ that converges to some $a \in A$. Then $$d(x_{n_k}, a) \le d(x_{n_k}, a_{n_k}) + d(a_{n_k}, a) < d(x_{n_k}, A) + 2^{-n_k} + d(a_{n_k}, a) \to 0,$$ hence $x_{n_k} \to a \in A$.