Let $A\subset\mathbb{R}^n$ be a compact and $x\in A$. Suppose that $x\_n$,$n\in N$ is a sequence in $A$ with the property that every convergent subsequence of $x\_n$ converges to $x$. Prove that the sequence $x\_n$ converges.

I try to argue by contradiction. ie there exist a $1$ and for all $n$ we can make distance of $x\_n$ and $x$ greater than $0$ equal to $1$. but how to go the further. please help me out.