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Let $f_n :[0,1] to Bbb{R}$ a sequence of uniformly bounded measurable functions with the property: $$int_0^1f_n(x)f_m(x)dx=0,forall m neq n$$

Prove that $frac{1}{N}sum_{n=1}^Nf_n(x) to 0$ almost everywhere in $[0,1]$.

I have already proven that:

$(1)$ $frac{1}{N^2}sum_{n=1}^{N^2}f_n(x) to 0$ almost everywhere.

$(2)$ $frac{1}{N+N^2}sum_{n=1}^{N+N^2}f_n(x) to 0$ almost everywhere.

$(3)$ $frac{1}{N}sum_{n=1}^{N}f_n(x)-frac{1}{N}sum_{n=1}^{N}f_{n+m}(x) to 0$ almost everywhere, $forall m in Bbb{N}$

Because of the fact that we have already a subsequence converging to $0$ almost everywhere i tried to show that $frac{1}{N}sum_{n=1}^{N}f_n(x)$ is a Cauchy sequence for almost every $x$.
But i did not manage anything.

Can someone give me a hint to solve this?

I do not want a full solution.

Thank you in advance.

The conditions (1) and that the $f_n$'s are uniformly bounded are sufficient.

That is, let $(a_n)_{n ge 1}$ be any sequence of reals such that $|a_n| le C$ for each $n$ and $frac{1}{N^2}sum_{n le N^2} a_n to 0$. Then $frac{1}{N}sum_{n le N} a_n to 0$. The reason this is true is that the squares occur frequently enough in the integers.

Suppose $M^2 le N le (M+1)^2$. Then $$|frac{a_1+dots+a_N}{N}| le |frac{a_1+dots+a_N}{M^2}| le |frac{a_1+dots+a_{M^2}}{M^2}|+|frac{a_{M^2+1}+dots+a_N}{M^2}|$$ $$ le |frac{a_1+dots+a_{M^2}}{M^2}|+frac{N-M^2}{M^2}C.$$ Now just let $M to infty$, noting that $frac{N-M^2}{M^2} le frac{2M+1}{M^2} to 0$.