Royden 8.2 - Criterion for convergence in $L^2(E)$

Question

I am trying to solve the following problem from Royden

16. Let $E$ be a measurable set, $\left\{f_{n}\right\}$ a sequence in $L^{2}(E)$ and $f$ belong to $L^{2}(E)$. Suppose $$ \lim _{n \rightarrow \infty} \int_{E} f_{n} \cdot f=\lim _{n \rightarrow \infty} \int_{E} f_{n}^{2}=\int_{E} f^{2} . $$ Show that $\left\{f_{n}\right\}$ converges strongly to $f$ in $L^{2}(E)$.

Here is my solution.

$$ \begin{aligned} &\lim _{n \rightarrow \infty}\left\|f_{n}\right\|_{2} \\ &=\lim _{n \rightarrow \infty}\left(\int_{E} f_{n}^{2}\right)^{1 / 2} \\ &=\left(\lim _{n \rightarrow \infty} \int_{E} f_{n}^{2}\right)^{1 / 2} \\ &=\left(\int_{E} f^{2}\right)^{1 / 2} \\ &=\|f\|_{2} \quad \text { So } f_{n} \rightarrow f \text { in } L^{2}(E) \end{aligned} $$

Why is this not correct?

The solution manual online gives this as the solution

16. By linearity of integration, we have $$ \int_{E}\left(f_{n}-f\right)^{2}=\int_{E} f_{n}^{2}-2 \cdot \int_{E} f \cdot f_{n}+\int_{E} f^{2} $$ Taking limits, we obtain $$ \lim _{n \rightarrow \infty} \int_{E}\left(f_{n}-f\right)^{2}=0 $$ Therefore $\left\{f_{n}\right\} \rightarrow f$ in $L^{2}(E)$.

EDIT

Answer 1

We say $f_n \to f$ in $L^{2}(E)$ if $\|f_n-f\|_2 \to 0$. Covergence of $\|f_n\|_2$ to $\|f\|_2$ is not enough for convergence in $L^{2}(E)$.