Prove that the limit of the inner product is equal to the inner product of the limits in $L^2$ I'm working on the following problem.

Let $\{f_n\},f,\{g_n\},g$ be functions from $[-\pi,\pi]$ to $\mathbb{C}$ that are all Riemann integrable. Prove that if $\{f_n\}$ converges in the mean to $f$ and $\{g_n\}$ converges in the mean to $g$ then $$\Vert f\Vert=\lim_{n\rightarrow\infty}\Vert f_n \Vert$$ and $$(f,g)=\lim_{n\rightarrow\infty}(f_n,g_n)$$ where $(f,g):=\int_{-\pi}^\pi f(x)\overline{g(x)}dx$ and $\Vert f \Vert:=\left[\int_{-\pi}^\pi |f(x)|^2dx\right]^{1/2}$

For the first one I have got something which seems too trivial...
$$(\lim_{n\rightarrow \infty}\Vert f_n\Vert)^2=\lim_{n\rightarrow\infty}\int_{-\pi}^\pi|f(x)|^2dx=(f(x))^2$$ and taking the square root of both sides gives the result. But I feel like I am missing some subtlety.
The second one I'm sure sure even how to get started.

Assuming you can use the fact that $(f,g)$ is an inner product, the second claim follows directly from the triangle inequality and the Cauchy-Schwarz inequality:
\begin{align}
|(f_n,g_n)-(f,g)|&=|(f_n-f,g_n-g)+(f_n,g)+(f,g_n)-2(f,g)|\\
&\leq |(f_n-f,g_n-g)|+|(f_n-f,g)|+|(f,g_n-g)|\\
&\leq ||f_n-f||\cdot ||g_n-g|| + ||f_n-f||\cdot ||g||+||f||\cdot ||g_n-g||\\
\end{align}
Which goes to zero since $||f_n-f||$ and $||g_n-g||$ go to zero. The claim follows.
The first claim is now trivial.
