Suppose that a sequence of continuous function $(f_n)$ converges point wise to $f$ on $[0,1]$.
Prove that if there exists a sequence $(x_n)$ in $[0,1]$ converging to $x^*$ in $[0,1]$ such that $(f_n(x_n))$ does not converge to $f(x^*)$, then $(f_n)$ is not uniformly convergent.