Sequences in Banach spaces I am very bad with proofs that ask you to show that "$\exists$ something..." because in most of them you have to explicitly show the something. The following question is one such. Any help will be greatly appreciated!
Let $\{u_n\}$ a sequence in a Banach space $X$. Suppose that $\sum_{n=1}^{\infty} \|u_n\| < \infty$. Prove that there exists some $x \in X$ such that
$$\lim_{n \to \infty} \sum_{k=1}^{n} u_k = x$$
Thank you very much in advance!
 A: Let $ m,n \in \mathbb{N} $ satisfy $ m \leq n $. By the Triangle Inequality for the norm $ \| \cdot \|_{X} $, we have
$$
\left\| \sum_{k=m}^{n} u_{k} \right\|_{X} \leq \sum_{k=m}^{n} \| u_{k} \|_{X}.
$$
As $ \displaystyle \sum_{k=1}^{\infty} \| u_{k} \|_{X} < \infty $, it follows from the Cauchy Criterion for Series that for every $ \epsilon > 0 $, there exists an $ N \in \mathbb{N} $ sufficiently large so that
$$
\forall m,n \in \mathbb{N}_{\geq N}: \quad m \leq n ~ \Longrightarrow ~ \sum_{k=m}^{n} \| u_{k} \|_{X} < \epsilon.
$$
Consequently,
$$
\forall m,n \in \mathbb{N}_{\geq N}: \quad m \leq n ~ \Longrightarrow ~ \left\| \sum_{k=m}^{n} u_{k} \right\|_{X} < \epsilon.
$$
We thus see that the sequence $ \displaystyle \left( \sum_{k=1}^{n} u_{k} \right)_{n \in \mathbb{N}} $ is Cauchy in $ X $. As $ X $ is a Banach space, every Cauchy sequence in $ X $ converges; in particular, the sequence $ \displaystyle \left( \sum_{k=1}^{n} u_{k} \right)_{n \in \mathbb{N}} $ converges to some $ x \in X $:
$$
\sum_{k=1}^{\infty} u_{k} ~ \stackrel{\text{def}}{=} ~ \lim_{n \to \infty} \sum_{k=1}^{n} u_{k} = x.
$$
