Consider a particular sequence $a_n \in \mathbb{C}$ with the particular property that $\forall n \in \mathbb{N}, |a_{n+1} - a_n| < C\lambda^n$, where $C = a_1 - a_0$ and $\lambda < 1$. My goal is to prove that $a_n$ is Cauchy.
So far I have the following argument: $$|a_m - a_n| = |a_m - a_{m-1} + a_{m-1} - a_{m-2} + ... + a_{n+1} - a_n| \leq |a_m - a_{m-1}| + |a_{m-1} - a_{m-2}| + ... + |a_{n+1} - a_n| \leq C\lambda^{m-1} + C\lambda^{m-2} + ... + C\lambda^n.$$
I also know that $C\lambda^n \rightarrow 0$, so that $ \forall \epsilon, m, n > 0,|C\lambda^n| < \frac{\epsilon}{m-n}$. How should I proceed from here?
For each $n,p>0$ we have as you wrote
$|a_{n+p}-a_n|\leq C\lambda^n\frac{1-\lambda^p}{1-\lambda}\leq \frac{C}{1-\lambda}\lambda^n$.
but $\lim_{n\to\infty}\lambda^n=0$
thus
$\forall \epsilon>0 \; \exists N>0 : \forall n>N \;\forall p>0 $
$|a_{n+p}-a_n|\leq \frac{C}{1-\lambda}\lambda^n<\epsilon$
$(a_n)$ is then a Cauchy sequence.