Let $(a_n)$ be a sequence such that $\lim_{N\rightarrow \infty} \sum_{n=1}^N |a_n - a_{n+1}| < \infty$. Show that $(a_n)$ is Cauchy.
Proof: We know $(x_n)$ is Cauchy if $\forall\ \epsilon > 0, \exists\ N \in \mathbb{N}$ such that $m, n \geq N \implies |x_{m} - x_{n}| < \epsilon$.
Since we have that $\lim_{N\rightarrow \infty} \sum_{n=1}^N |a_n - a_{n+1}| < \infty$, and is the limit of the sum of absolute values (hence always positive and cannot diverge to $-\infty$), $\lim_{N\rightarrow \infty} \sum_{n=1}^N |a_n - a_{n+1}| = L$, for some $L \in \mathbb{R}$.
So we have that the limit of our sequence $|a_1 - a_{2}|, |a_1 - a_{2}| + |a_2 - a_{3}|\ + \dots$, converges.
Let $\epsilon > 0$. Take $x_n = \sum_{k=1}^n |a_k - a_{k+1}|$. Hence $\exists\ N \in \mathbb{N}$ such that $k \geq M \implies |x_k - x| < \frac{\epsilon}{2}$.
So $m , n \geq M \implies |x_m - x_n| = |(x_m - x) - (x_n - x)| \leq |x_m - x| + |x_n - x| < \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon$
So our sequence is Cauchy.
See comments, misread the question.