Claim. Given a sequence $\{p_n\}_{n=0}^\infty$ such that $p_n \to p$ and a constant $0 \leq k < 1$ such that $|p_n - p| \leq k |p_{n-1} - p|$ for every integer $n \geq 1,$ prove that for every integer $n \geq 1$ $$|p_n - p| \leq \frac{k}{1-k} |p_n - p_{n-1}|.$$
I have been thinking about this problem for the entire weekend, and I am still not sure how to approach it. Unfortunately, the Triangle Inequality does not seem to be very helpful. I would greatly appreciate any hints or tips that could be provided.
Since
$$ |p_n - p| \leq k|p_{n-1}-p| \leq k|p_{n-1}-p_n| + k|p_n-p| $$
we get
$$ (1-k)|p_n-p| \leq k|p_n - p_{n-1}| $$
and dividing by $(1-k)$ which is positive,
$$ |p_n-p| \leq \frac{k}{1-k}|p_n-p_{n-1}|. $$
Note that convergence is not necessary since it is a consequence of the former bound:
$$ |p_n -p| \leq k|p_{n-1}-p| \leq \dots \leq k^{n}|p_0-p| \xrightarrow{n \to \infty} 0. $$