Prove that $x_{n} \rightarrow x$

Question

Suppose that we are given a point $x$ and a sequence ${(x_{n})}$ in a metric space $M$ and suppose that $f(x_{n}) \rightarrow f(x)$ for every continuous real valued function $f$ on $M$. Prove that $x_{n} \rightarrow x$

I am not able to understand how I am suppose to start with the problem.

Kindly guide me through !

Thanks!

Answer 1

Call the metric $d$. Verify that $d(\_,x):M\rightarrow \mathbb{R}$ is a continuous function. Apply your assumption to $d(\_,x)$. Conclude that $d(x_{n},x)\rightarrow 0$.