Let {$S_n$} be a sequence of real numbers, and for each n in set of positive integers ,
let $S_n = a_1+a_2+a_3+...+a_n$
$t_n =|a_1|+|a_2|+|a_3|+|a_4|+...|a_n|$.
Prove that {$t_n$} is a cauchy sequence then so is {$S_n$}.
The proof goes like:-
{$t_n$} is a Cauchy seq.
implies that
for every $\epsilon>0$, there exists a positive integer N such that $n>m\ge N$ implies that $|t_n-t_m|< \epsilon$.
implies that
for every $\epsilon >0$, there exists a positive integer $N$ such that $n>m \ge N$ implies that
$| |a_m|+ |a_{m+1}|+ ...+|a_n| | < \epsilon$
[From where these terms come from? How are these terms obtained? I want some steps to clarify how these terms come and an example will probably help.]
Let $(s_n)$ be the sequence of partial sums of the infinite series $\sum a_n$ associated to the sequence $(a_n)$. That is, \begin{align*} s_1 & =a_1 \\ s_2 & = a_1 + a_2 \\ \vdots & = \cdots \\ s_n & = a_1 + \cdots + a_n \\ \vdots & \end{align*} $(t_n)$ is the sequence of partial sum of the series $\sum |a_n|$.
Now, let $(t_n)$ be Cauchy. Then for any $ \epsilon >0$ we can find an integer $N=N(\epsilon)$, that is, dependent on $\epsilon $ , so that, for all $n,m \geq N$, we have that ($m>n$): \begin{align*} |t_m - t_n|&= \left||a_1|+...+ |a_m| - (|a_1|+...+|a_n|) \right| \\ & = \left||a_{n+1}| +...+|a_m|\right| < \epsilon \end{align*}
Now, by the triangle inequality, it holds that: $$ \left| a_{n+1} + ... + a_m \right| \leq \left||a_{n+1}| +...+|a_m|\right|.$$
Thus, our result follows immediately by choosing for $(s_n)$ the same $N(\epsilon)$ previously chosen for $(t_n)$.
What we have shown is that absolute convergence implies convergence.