convergence of sequences in $T_4$ space

Question

I have recently been asked this

For a normal Hausdorff space we are to show a sequence $x_n$ converges to $y$ iff for all continuous real-valued functions $f(x_n)$ converges to $f(y)$

One direction is easy enough but how to deduce convergence of sequence from convergence of functions?

Thanks for the help

Answer 1

HINT: Suppose that $\langle x_n:n\in\Bbb N\rangle$ does not converge to $y$. Then $y$ has an open nbhd $U$ such that $M=\{n\in\Bbb N:x_n\notin U\}$ is infinite; why? Now use the fact that your space is completely regular, by applying the definition of complete regularity to the point $y$ and the closed set $X\setminus U$, and show that the resulting function does what you want.