Let $\Gamma$ be set and $p \in [1,\infty[$. Show that $ \ell_{p}(\Gamma)=\{(x_i)_{i \in \Gamma}: x_i \in \mathbb{R} \text{ and } \sum_{i \in \Gamma}|x_i|^p < \infty\}$ is complete. $\textbf{Trought:}$ Let $(x_n)_{n \in \mathbb{N}}$ be sequence of cauchy in $\textit{$\ell_{p}(\Gamma)$}$ then $\forall \epsilon >0 \exists n_0 \in \mathbb{N}$ such that if $n,m>n_0$: $$||x_m-x_m||_{l_p}=( \sum_{i \in \Gamma}|x_i^m-x_i^n|^p )^\frac{1}{p}< \epsilon$$ fixed $j \in \Gamma$ if $n,m>n_0$ $$|x_j^m-x_j^n|< \epsilon$$
we have a sequence of cauchy in $\mathbb{R}$ then $x_j^m \to x_j$. with this limit define the sequence $x=(a_j)_{j \in \Gamma}$. Now we have to show that $x \in \textit{$\ell_{p}(\Gamma)$}$ and $x_n \to x$.
To show that $x_n \in \ell_{p}(\Gamma)$ i want to use that sequence of cauchy is limited. Let $a>0$ such that $||x_k||\leq a \forall k \in \mathbb{N}$
Now I do not know what to do, the idea is to do $(\sum|x_i^k|^p)^\frac{1}{p} \leq a$ with $i$ in a finite subset of $\Gamma$, to send k to infinity and then back to $\Gamma$. Can you formalize this idea?
Do you know what a sum like $\sum_{i \in I} a_i$ stands for when$I$ is an arbitrary set and $a_i \geq 0$?. By definition it is the sup of all finite sums of $a_i$'s. So to show that $\sum |x_i^{k}|^{p} <\infty$ you only have to show that $\sum_{i\in F} |x_i^{k}|^{p} $ is bounded when $F$ ranges over all finite subsets $F$ of $\Gamma$. This sum is bounded by the supremum of the norms of $x_n$'s, so we are done. Use a similar argument to show that $x_n \to x$ in $\ell_{p}(\Gamma)$.