Proving convergence of $(a_n)$ given $|a_n-a_{n-1}|<2^{-n}$ for all $n \in \mathbb{N}$

Question

I'm trying to prove the convergence of $(a_n)$ given $|a_n-a_{n-1}|<2^{-n}$ for all $n \in \mathbb{N}$. I think this is related to Cauchy's Criterion and how one may choose $n$ such that $2^{-n}<\varepsilon$ for any $\varepsilon > 0$, but I am not sure how to proceed from here. Thank you in advance for the help!

Answer 1

\begin{align*} |a_{m}-a_{n}|&=|a_{m}-a_{m-1}+\cdots+a_{n+1}-a_{n}|\\ &\leq|a_{m}-a_{m-1}|+\cdots+|a_{n+1}-a_{n}|\\ &<\dfrac{1}{2^{m+1}}+\cdots+\dfrac{1}{2^{n+1}}\\ &<\dfrac{1}{2^{n+1}}+\cdots\\ &=\dfrac{1}{2^{n}} \end{align*} for $m>n$, can you finish from here?