IMC 1995 Problem 12

Suppose that $(f_{n})_{n=1}^{\infty}$ is a sequence of continuous functions on the interval $[0,1]$ such that $$\int_{0}^{1}f_{m}(x)f_{n}(x) dx= \begin{cases} 1& \text{if} n=m 0 & \text{if} n\ne m \end{cases}$$ and $\sup\{|f_{n}(x)|: x\in [0,1] \text{and} n=1,2,\dots\}< \infty$ . Show that there exists no subsequence $(f_{n_{k}})$ of $(f_{n})$ such that $\lim_{k\to \infty}f_{n_{k}}(x)$ exist for all $x\in [0,1]$ .