Suppose that $(x_n)$ is a sequence satisfying $|x_{n+1}-x_n| \leq q|x_n-x_{n-1}|$ for all $n \in \mathbb{N^+}$.Prove that $(x_n)$ converges Let $q$ be a real number satisfying $0<q<1$. Suppose that $(x_n)$ is a sequence satisfying $$|x_{n+1}-x_n| \leq q|x_n-x_{n-1}|$$ for all $n \in \mathbb{N^+}$. Prove that $(x_n)$ is a convergent sequence. I try to show that the sequence is Cauchy but I stuck at finding the $M$. Can anyone help me ? 

Hint: note that for any $N$ and $n>N$, we have $|x_{n+1}-x_n|\leq q|x_n-x_{n-1}|\cdot q^{N-1}|x_2-x_1|$. This can be proven by induction on $N$. If we have $n>m>N$, then how can we bound $|x_n-x_m|$?
