show that sequence $(a_n)$ is convergent

Question

Let $(a_n)$ be a sequence of real numbers. If $$|a_{n+1} − a_n | ≤ \frac{1}{2}\ |a_n − a_{n−1} |, \qquad \forall n \in \{2, 3, \dots\}$$ then show that $(a_n)$ is convergent.

Answer 1

Hint: $$|a_{n+1}-a_n|\le\frac{1}{2^{n-1}}|a_2-a_1|$$

Answer 2

First step: $$|a_{n+1} - a_n| \leq \frac{1}{2^{n-1}}|a_2 - a_1|$$

Second step:

$$|a_n - a_m| \leq \sum_{k=m}^{n-1}|a_{k+1} - a_k| \leq \sum_{k=m}^{n-1}\frac{1}{2^{k-1}}|a_2 - a_1| \leq |a_2 - a_1|\sum_{k=m}^{+\infty}\frac{1}{2^{k-1}} = \dfrac{|a_2 - a_1|}{2^{m-2}}$$

Now conclude $a_n$ is a Cauchy sequence, so it converges