Let $f_n, f$ be real-valued functions on $[0,1]$ with $f$ continuous. Suppose that for all convergent sequences $\{x_n : n \geq 1\} \subseteq [0,1]$ with $x = \lim_{n \to \infty} x_n$ one has
$$\lim_{n \to \infty} f_n(x_n) = f(x).$$
Show that $f_n$ converges to $f$ uniformly.