If $\{a_n\}$ is a cauchy-sequence. Show that $$\lim_{n\to \infty}|a_n-a_{n+1}|=0$$
It makes intuitive sense, I tried by substituting $b_n := a_n-a_{n+1}$ , but couldn’t get an answer.
I should note that we are dealing with the set of rational numbers $\Bbb Q$.
Suppose that $\{a_n\}$ is a Cauchy sequence. Let $\varepsilon > 0$. Then, by definition, there exists $N \in \mathbb N$ such that $$|a_n - a_m| \leq \varepsilon$$ for all $n,m \geq N$. Letting $m = n + 1$ yields that $$|a_n - a_{n+1}| \leq \varepsilon,$$ which is precisely what we wanted to show.