Let $(a_n)$ be a sequence of real numbers and for each $n\in N$, let $f_n=a_1+a_2+a_3+...+a_n$ and $g_n$=$|a_1|+|a_2|+...+|a_n|$. Prove that if $(g_n)$ is a Cauchy sequence, then so is $(f_n)$.
My attempt: Since $(g_n)$ is a Cauchy sequence, we have,
$||a_1|+|a_2|+|a_3|+...|a_n||<\epsilon$ for all $n\geq m$
This implies $|a_1+a_2+...a_n|<\epsilon$ for all $n\geq m$
Thus $(f_n)$ is a Cauchy sequence.
Note that $$|f_n - f_m| = |a_{m+1} + \dots + a_n| \le |a_{m+1}| + \dots + |a_n| = |g_n - g_{m}| < \varepsilon$$