Proving that the sequence space $l^1$ is complete

Question

I don't know how to figure this out :

Answer 1

You know that $\{x_i^{n}\}$ is Cauchy so it is bounded, namely, $\exists M > 0$ such $\sum\limits_{i=1}^{\infty}{|x_i^{n}|} \leq M$. Thus, $\forall k \in \mathbb{N}$, $\sum\limits_{i=1}^{k}{ |x_i^{n}|} \leq M$. Taking the limit as $n \to \infty$ we get that $\sum\limits_{i=1}^{k}{|x_i|} \leq M$. Since this holds for all $k \in \mathbb{N}$, letting $k \to \infty$ we get that $\sum\limits_{i=1}^{\infty}{|x_i|} \leq M$. Thus, $x=\{x_i\} \in \ell^1(\mathbb{N})$.