Strong convergence in L2(E) if this limit and integral exchange

Theorem: (Royden Real Analysis 4th ed. Section 8.2 Problem 16)

Suppose that E is a measurable set and every function in sight lives in $L^2(E)$. Further suppose

$$lim{\int }_E{f}_{n}f\to {\int }_E{f}^2=lim{\int }_E{f}_n^2$$

Then $\{f_n\}\to f$ strongly in $L^2(E)$.

Proof:

$${\int }_E(f_n-f{)}^2={\int }_E{f}_n^2-2{\int }_{E}f{f}_n+{\int }_E{f}^2$$

Which has limit zero by the assumptions of the problem.