Theorem: A normed vector space $(V,||\circ||)$ is a banach space if and only if for every sequence $x_n$ in $V$ with the property that $\sum ||x_n||<\infty$ we have $\sum x_n < \infty$.
Question: Let $L=\{x=(x_n) \subset \mathbb R: \sum |x_n|<\infty\}$ and $||x||=\sum |x_n|,\forall x \in L$. Prove that $(L,||\circ||)$ is a banach space.
So I would guess that I should use the above theorem to solve this problem. So suppose we have $v_n \in L$ and $\sum ||v_n|| < \infty $ then we need to show that $\sum v_n < \infty$ but what does $\sum v_n < \infty$ mean where each $v_n$ is a sequence. How do we sum a series of sequences?
So I tried to use the theorem above but it got very complicated and instead I've tried to use the completeness of $\mathbb R$. Suppose that $l_n$ is a Cauchy sequence in $L$. Then we have $$\forall \epsilon>0,\exists N \in \mathbb N \ such \ that\ n,m>N \Rightarrow ||l_n-l_m||<\epsilon$$ So by the reverse triangle inequality we have $$| \ ||l_n||-||l_m||\ | \leq ||l_n-l_m||<\epsilon$$ which implies $||l_n||$ is cauchy in $\mathbb R$ and therefore converges to some $k \in \mathbb R$.
Let $(k_n)=K$ be a sequence in $\mathbb R$ such that $\sum |k_n|=k$. Clearly $K \in L$ so we just need to show that $l_n \rightarrow K$.
Is this correct so far? If anybody could show me how the last step is done that would be great. I've been trying for a while but can't seem to do it.