Prove that sequence $\frac{a_{n+1}}{a_n} \le r$ converges to $0$

Question

This question is about the following:

Let $(a_n)_{n=0}^{\infty}$ be the sequence of positive numbers.

Let $r \in \mathbb{R} \, \wedge \, 0 < r < 1$

$ \forall n \in \mathbb{N}_0 : \frac{a_{n+1}}{a_n} \le r$

Prove that $\lim\limits_{n \to \infty} a_n = 0$

The sequence $a_n$ is bounded from below by $0$ because of its definition.

But I was not able to come up with an proof that the sequence $a_n$ converges to $0$.

Answer 1

Here's a nice trick:

$$a_n=\frac{a_n}{a_{n-1}}\frac{a_{n-1}}{a_{n-2}}\cdots\frac{a_2}{a_1}a_1.$$

Can you finish the rest from here?