Let $f_n, f \in L^2$ and suppose that $\lim_{n \to \infty}\int f_nf=\lim_{n \to \infty}\int f_n^2=\int f^2$. Show $f_n \to f$ in $L^p$

Question

Let $f_n, f \in L^2$ and suppose that $\lim_{n \to \infty}\int f_nf=\lim_{n \to \infty}\int f_n^2=\int f^2$. Show $f_n \to f$ in $L^p$

My "idea"

$\int (f_n-f)^2=\int(f_n^2 -2f_nf+f^2)$ then by our limits we know that for large enough $n$ $||f_n-f||_2<\epsilon$

This just seems too easy.

Answer 1

Answer for $p=2$: $\int (f_n-f)^{2}=\int f_n^{2}+\int f^{2}-2\int f_n f \to \int f^{2} -2\int f^{2} +\int f^{2}=0$.