$g \in L^p(\Omega)$ Let $\Omega$ is a measurable set of $\mathbb{R}^n$. Given $g_m \in L^p(\Omega)$ such that $\sum_{m=1}^{\infty} \left \| g_m \right \|_{L^p} < \infty$. Let $g(x)= \sum_{m=1}^{\infty} g_m(x)$. Is the following true or false? $g \in L^p(\Omega)$
I'm beginning at real analysis, can you give me some sources of exercises like that?

All $L^p$ spaces are Banach spaces. This means that they are complete (i.e. all convergent sequences converge to a function within the space). 
Note that $g_m$ is a sequence of functions in $L^p(\Omega)$. Since it is a Banach space, this means the limit $g_m\rightarrow g$ is also in the space. This extends to the sum, as we are dealing with a sequence of partial sums. 
Here are some good exercises:
$1)$ Show the space of continuous functions with compact support $C_c(\mathbb{R})$ is not complete.
hint: find a sequence of functions which each are compactly supported but the limit function only vanishes at infinity, so it is not a member of the space
$2)$ Show that $L^{\infty}(\mathbb{R})$ is complete. 
