By definition I know that a contractive sequence is called contracting if there exists $l\in (0,1)$ such that for $n>N$:
$$|x_{n+2}-x_{n+1}|\le l |x_{n+1}-x_n|;\forall n \in\mathbb{N}$$
There is an inequality $|x_{n+1}-x_n| \le l^{n-1} |x_2-x_1|$ which seems related to the contractive sequences that is written in my lecture notes but I am not sure how this holds true. Can anyone guide me on some steps or show me how this inequality holds true? It will be really helpful.
$$|x_{n+1}-x_n| \le l|x_n-x_{n-1}| \le l^2|x_{n-1}-x_{n-2}| \le \cdots \le l^{n-1} |x_2-x_1|.$$