Bound on the distance between two consecutive terms of a sequence guaranteeing the Cauchy condition

Question

Let $0 < a < 1$ and let $(a_n)$ be a sequence in $\mathbb{R}$ such that $|a_{n+1} - a_n| < a^n.$ I want to show that $(a_n)$ is Cauchy.

My idea was to bound

$$ |a_{n+k} - a_n| \leq \sum_{i=0}^{k-1} |a_{n+k-i} - a_{n+k-i-1}| \leq ka^n,$$

but this clearly isn't tight enough (since $k$ can be arbitrarily large).

Any input would be greatly appreciated!

Answer 1

This is not a correct estimate.

You should have:

$|a_{n+k}-a_n| \leq \sum\limits_{i=0}^{k-1}a^{n+i}$. Since the geometric series converges this sum is "small".

Answer 2

You have to use the following inequalities

$|a_{n+k}-a_{n+k-1}|<a^{n+k-1}$

$|a_{n+k-1}-a_{n+k-2}|<a^{n+k-2}$

............

$|a_{n+1}-a_{n}|<a^{n}$

and then take the sum. This will work.