Question Regarding the convergence of $\sum_{n=1}^\infty |x_n| < \infty$. I am currently reading a chapter on normed vector spaces in N.L. Corothers' Real Analysis textbook and came across the following in a closing paragraph:

$l_1$ space consists of sequences {$x_n$} such that $\sum_{n=1}^\infty |x_n| < \infty$. Therefore, as $n \to \infty$ we have that $x_n \to 0$.

How is the second sentence possible? Is it because the sum of the $a_n$'s  is less than infinity? Maybe this is in reference to some convergence theorem I haven't met/forgotten. In any case, I am seeking some clarity.
 A: For
$$L = \sum_{n=1}^{+\infty} |x_n|$$
you have $$\lim_{n \rightarrow +\infty} |x_n| = \lim_{n \rightarrow +\infty} \left( \sum_{k=1}^{n} |x_k| - \sum_{k=1}^{n-1} |x_k|\right) = L-L = 0$$
A: I'll try to provide a more intuitive insight here.
Let $x_n$ be a sequence.That means $x_n$ is a sequence of $x_1,x_2,x_3,...x_n,x_{n+1},...$. The sum $\sum_{n=1}^\infty |x_n|$ represents the sum of the members of that sequence i.e. $$x_1+x_2+x_3+...+x_n+x_{n+1}+...$$
If $lim_{n \to \infty}|x_n|\neq0$ that would mean that you would always be adding some values of $x_n>0$ what would get you to infinity eventually.
Let's say $lim_{n \to \infty}|x_n|=\frac{1}{2}\neq0$, that would mean that after very large values of $n$ the term $x_n$ still wouldn't get smaller than $\frac{1}{2}$ (if the sequence $x_n$ is decreasing, or it wouldn't get larger than $\frac{1}{2}$ if the sequence is increasing) because the limit is a bound of the sequence. And if you keep adding those terms, let's say $x_{1000}=0.55$, $x_{1500}=0.54$, $x_{10000}=0.51$ etc., you would get  $x_{1000} + x_{1500} + x_{10000}=1.6$, and that is only the sum of three arbitrarily chosen values of $x_n$. And as you can see for even larger $n$s the values of $x_n$ would still be $x_n \approx 0.5$, and it should be obvious from this point that the sum just keeps increasing and there is nothing preventing the sum to eventually get to infinity.
$lim_{n \to \infty}|x_n| = 0$ tells us that for large values of $n$ you would get such small values for $x_n$ which would esentially stop increasing the sum at one point (e.g. $x_n=00000.1$, $x_{n+100000}=0.00000000001$ etc.). It basically proves that there is actually something preventing the sum to get to infinity, and therefore it is a neccessary condition for a series to converge.
