About Banach Spaces And Absolute Convergence Of Series
How to prove the following two assertions:
If in a normed space $X$, absolute convergence of any series always implies convergence of that series, then $X$ is a Banach space.
In a Banach space, absolute convergence of any series always implies convergence of that series.
Take a Cauchy sequence $x_k$. Then you can find a subsequence $n_k$ such that $|x_{n_{k+1}}-x_{n_k}| < 2^{-k}$. Let $y_k = x_{n_{k+1}}-x_{n_k}$. Then $\sum y_k$ is absolutely convergent and, by hypothesys, it converges. But $\sum_{k=1}^N y_k = x_{n_N} - x_{n_1}$ and hence $x_k$ has a convergent subsequence. But since $x_k$ is Cauchy, the whole sequence is convergent.