Let $f_n \in L^2[0,1]$ and $\|f_n\|_2=1$ and suppose $\lim_{m,n \to \infty}\|f_m+f_n\|_2=2$, prove that there is $f \in L^2$ s.t $f_n \to f$ in $L^2$

Question

Let $f_n \in L^2[0,1]$ and $\|f_n\|_2=1$ and suppose $\lim_{m,n \to \infty}\|f_m+f_n\|_2=2$, prove that there is $f \in L^2$ s.t $f_n \to f$ in $L^2$

I think I need to show the sequence is cauchy, but I am not sure how to do that. I would write what I tried but I am not even sure what to try here.

Answer 1

Hint:

$$\int (f_m+f_n)^2 = \int f_m^2+2\int f_mf_n +\int f_n^2 = 2 + 2\int f_mf_n.$$

This tells us that $\int f_mf_n\to 1.$ Now look at $\int (f_m-f_n)^2.$