Why isn't the completion of $C^0$ wrt. the $L^2$ norm a space of sequences instead of a space of functions? We know that $L^2(\Omega)$ can be defined as the completion of $C^0(\Omega)$ with respect to the norm $$\left(\int_\Omega |u|^2\right)^{\frac 12}.$$
But strictly speaking, $L^2(\Omega)$ is a space of sequences since that is how the completion of a metric space is defined. So why is it people say $f \in L^2(\Omega)$ to mean an actual function (with the a.e. caveat)instead of a sequence? 

We can identify a sequence $u=(u_n)$ with $u$ itself if $(u_n)$ is the constant sequence (eg. $u_i = u_1$ for all $i$), which is the case for the subspace $C^0(\Omega)$. But for a general element of $L^2(\Omega)$, we cannot do this.
Because $L^2$ can also be defined as the space of square-integrable functions (up to measure zero), and one can prove both definitions are equivalent. It's not any more economical to represent $L^2$ as a space of sequences than to do so for $\mathbb{R}$: they're each only spaces of sequences up to a rather large equivalence relation, so elements don't have very unique representatives.
Because $L^2$ can also be defined as the space of square-integrable functions (up to measure zero), and one can prove both definitions are equivalent. It's not any more economical to represent $L^2$ as a space of sequences than to do so for $\mathbb{R}$: they're each only spaces of sequences up to a rather large equivalence relation, so elements don't have very unique representatives.