Let’s say I have $f_1,f_2,..,f_n \in C^0([0,1], \mathbb{R})$, such that : $$\forall i \ne j \langle f_i, f_j \rangle = 0 \text{ and } \forall i \| f_i \|= 1$$
Where the scalar product is : $\langle f,g \rangle = \int_0^1 fg$.
Then I need to prove that :
$$\forall n \exists i, \geq 1, \sum_{k=1}^n (\int_{(k-1)/n}^{k/n} f_i)^2 \leq 1/n$$
The way I prove this inequality is simply by using Cauchy-Schwarz : $$(\int_{(k-1)/n}^{k/n} f_i)^2 \leq 1/n \int_{(k-1)/n}^{k/n} (f_i)^2$$
Then just sum, and using the fact that : $\int_0^1 f_i^2 =1$ we have the desired inequality.
The problem is that my proof shows the inequality is true for every $i$ and moreover I never use the fact that :$ \langle f_i, f_j \rangle = 0$, so isn’t there something wrong with what I’ve done ?
Thank you !
There is nothing wrong with what you have done. Lot of unnecessary hypothesis in the question.