Suppose $f_{n} \stackrel{L_{2}}{\rightarrow} f$,$g_{n} \stackrel{L_{2}}{\rightarrow} g$. Prove that $ \langle f_n,g_n \rangle\to\langle f,g\rangle $

Theorem: Suppose $f_{n} \stackrel{L_{2}}{\rightarrow} f$ , $g_{n} \stackrel{L_{2}}{\rightarrow} g$ . Prove that $ \langle f_n,g_n \rangle \to \langle f , g\rangle $.

Proof from lecture notes:
$\left\langle f_{n}, g_{n}\right\rangle-\langle f, g\rangle=\left\langle f_{n}, g_{n}\right\rangle-\left\langle f, g_{n}\right\rangle+\left\langle f, g_{n}\right\rangle-\langle f, g\rangle=\left\langle f_{n}-f, g_{n}\right\rangle+\left\langle f, g_{n}-g\right\rangle$
And now, buy Cauchy-Schwartz $\left|\left\langle f, g_{n}-g\right\rangle\right| \leq\|f\|\left\|g_{n}-g\right\| \rightarrow 0$
Now, using the theorem " If $g_{n} \stackrel{L_{2}}{\rightarrow} g$ then $\left\|g_{n}\right\| \rightarrow\|g\|$ " , we get again by Cauchy-Schwartz
$\left|\left\langle f_{n}-f, g_{n}\right\rangle\right| \leq\left\|g_{n}\right\|\left\|f_{n}-f\right\| \rightarrow\|g\| \cdot 0=0$. Hence $\left\langle f_{n}, g_{n}\right\rangle-\langle f, g\rangle \rightarrow 0$ , as wanted. $ \square $

I tried proving the theorem above on my own first, as follows: $\left\langle f_{n}, g_{n}\right\rangle-\langle f, g\rangle = \left\langle f_{n}, g_{n}\right\rangle-\left\langle f, g_{n}\right\rangle+\left\langle f, g_{n}\right\rangle-\langle f, g\rangle= \left\langle f_{n}-f, g_{n}\right\rangle + \left\langle f, g_{n}\right\rangle-\langle f, g\rangle = \left\langle f_{n}-f, g_{n}\right\rangle + \overline{\left\langle g_{n}, f\right\rangle}-\overline{\langle g, f\rangle} $ But I didn't know how to proceed. Essentially, I don't see how they obtained $ \left\langle f, g_{n}-g\right\rangle $ from the part where it is written "... = $\left\langle f_{n}-f, g_{n}\right\rangle+\left\langle f, g_{n}-g\right\rangle $ ". I understand if the vector space I'm working in is over the field $ \mathbb{R} $, but what if the field is $ \mathbb{C} $? In that case I don't understand how to proceed from the point I've stopped.

Thanks in advance!

Notes:
I was given the following theorems:
$ \forall f,g \in R(\mathbb{T}) $ and $ \forall \lambda \in \mathbb{C} $.
1.$\langle g, f\rangle=\overline{\langle f, g\rangle} $
2.$\left\langle f_{1}+f_{2}, g\right\rangle=\left\langle f_{1}, g\right\rangle+\left\langle f_{2}, g\right\rangle$
3.$\langle\lambda f, g\rangle=\lambda\langle f, g\rangle $
4. Cauchy-Schwartz inequality: $|\langle f, g\rangle| \leq\|f\|_{2} \cdot\|g\|_{2}$
5. Triangle inequality: $\|f+g\|_{2} \leq\|f\|_{2}+\|g\|_{2}$

$ R(\mathbb{T}) $ is the set of functions $ \mathbb{R} \to \mathbb{C} $ that are periodical with a period of $ 2 \pi $ and integrable on $ [0, 2\pi ] $

$$\langle f, g+h \rangle$$ $$=\overline {\langle g+h, f \rangle}$$ $$=\overline {\langle g, f \rangle+ \langle h, f \rangle}$$ $$=\overline {\langle g, f \rangle}+\overline { \langle h, f \rangle}$$ $$=\langle f, g \rangle+\langle f, h \rangle$$