Let ${a_n}$ be a sequence such that there exists an M > 0 such that for all n ∈ N one has $|a_{n+1} − a_n|$ ≤ M/$2^n$
Prove that ${a_n}$ is a Cauchy sequence.
My attempt: I tried to use the triangle inequality to prove that for all m$>$n, $|a_m - a_n|$<$\epsilon$ but I am unable to get anything useful out of the inequalities I'm getting.
By induction and the triangle inequality, $|a_{n+k}-a_n|\le\frac{M}{2^n}\sum_{j=1}^k 2^{1-j}\le\frac{M}{2^{n-1}}$. For $\epsilon>0$, ensure $|a_m-a_n|<\epsilon$ for all $m,\,n\ge N$ by taking $N=\big\lceil\log_2\frac{2M}{\epsilon}\big\rceil$.