Real Analysis. Hello, im taking a Real Analysis course and I'm trying to finish this proof given as an exercise.
Question: Let $(X,||\cdot||)$ be a normed space. Let $x_n \subseteq X, a_n \subseteq \mathbb{R}$ be sequences, asuch that $\sum_{n=1} ^{\infty} ||x_n||$ converges. Suppose that $\limsup_{n\to\infty}|a_n|<\infty$
PROVE that $\sum_{n=1} ^{\infty} a_nx_n$ converges
My attempt: I know about the Cauchy convergence criterion, the definition we studied in class was: $\sum_{n=1} ^{\infty} a_n$ converges if and only if $||\sum_{k=n} ^{m} a_k|| < \epsilon$, given an $\epsilon>0$ if $\exists N \in \mathbb{N}$ such taht $\forall m,n \geq N$ $||\sum_{k=n} ^{m} a_k|| < \epsilon$.
Thus, usting the triangular inequality, I have that $||\sum_{k=n} ^{m} a_kx_k|| \leq \sum_{k=n} ^{m} ||a_kx_k||$.
Now, by properties of the Norm $||\sum_{k=n} ^{m} a_kx_k|| \leq \sum_{k=n} ^{m} |a_k|||x_k||$.
Now, my professor said that this property might be useful: " If $\alpha =\limsup_{n\to\infty}|b_n|$ and given an $\epsilon >0$ there are only finite "n´s" such that $b_n>\alpha + \epsilon$. However, I'm not too sure on how to use this...
I know that I need to in some sense "bound" $\sum_{k=n} ^{m} ||x_n||< \epsilon_1$, for some $\epsilon_1$....
Please any help is useful... Thanks in advance !!