Prove if $\vert a_{n+1}-a_n\vert\leq \ q\vert a_n - a_{n-1} \vert$ then $(a_n)_{n\in\mathbb{N}}$ is a Cauchy sequence

Question

I’m tasked with proving the following:

For $n_0 \in \mathbb{N}$ and $q\in \mathbb{R}$ with $0<q<1$ and $n \geq n_0$, let $(a_n)_{n\in\mathbb{N}}$ be a sequence in $\mathbb{R}$. If $$\vert a_{n+1}-a_n\vert\leq q\vert a_n - a_{n-1} \vert$$ then $(a_n)_{n\in\mathbb{N}}$ is a Cauchy sequence.

I thought it may be possible by trying to "form" it in such a way that I can use the ratio test, but it doesn't really work it seems. Alternatively I thought it's sufficient to prove that the sequence converges since that would automatically mean that it's a Cauchy sequence, but it seems to me that there is to little to work with. Any help would be appreciated.

Answer 1

As $\forall n>n_o $ $$\left | a_{n+1} -a_n\right | \le q \left | a_{n} -a_{n-1}\right | $$ Let$ n =n_o+1$ $$\left | a_{n_{o}+2} -a_{n_o+1}\right | \le q \left | a_{n_{o}+1} -a_{n_{o}}\right | $$ $n=n_o+2$ $$\left | a_{n_o+3} -a_{n_{o+2}}\right | \le q \left | a_{n_o+2} -a_{n_o+1}\right | \le q^2 \left | a_{n_o+1} -a_{n_{o}}\right | $$ Now generalizing give

$$\left | a_{n+1} -a_{n}\right | \le q^{n-n_o} \left | a_{n_o+1} -a_{n_o}\right | $$ Now let $ \lim_{n \to \infty}$ To get $\lim_{n \to \infty} \left | a_n-a_{n-1} \right | =0$