Limit of Function Guaranteed by Sequence Properties

Let $S\subset\mathbb{R}$, and $c$ a cluster point of $S$.

Define $f:S\to\mathbb{R}$, and suppose that for every sequence ${(x_n)_{n=1}^{\infty}}$ in $S\setminus \{c\}$ such that ${\lim\limits_{n\to\infty} x_n=c}$ the sequence $(f(x_n))_{n=1}^\infty$ is convergent.

Prove that $\lim\limits_{x\to{c}}f(x)$ exists.

Thanks for any help, I worked on this one for a while and got nowhere.