I am attempting to prove the Riesz-Fischer theorem, which asserts that for an open, connected, and bounded subset $\Omega$ of $\mathbb{R}^n$, the space $L_2(\Omega)$ is complete, i.e., a Banach space, equipped with the norm $$ \left\| f\right\|_{L_2} = \left( \int_\Omega f^2(x) \text{d}x \right)^{\frac{1}{2}}.$$
While I have successfully proven the theorem for the special case of $L_2([a, b])$, I am uncertain about extending this result to the more general case outlined above. My intuition suggests that the monotone convergence theorem and/or the dominated convergence theorem might be involved, but I am unsure about the specific steps. Can anyone provide guidance on bridging this gap in my proof?