Is $(a_n)_n$ with $|a_{n+2}-a_{n+1}| ≤ q|a_{n+1}-a_n|$ a Cauchy sequence? Let $0 < q < 1$ and $(a_n)_n$ be a squence with $|a_{n+2}-a_{n+1}| ≤ q|a_{n+1}-a_n|$ for all $n ∈ ℕ$. 
I need to show that this is a Cauchy sequence. I'm not sure how to start this one, as we just had cauchy sequences in lecture.
Can I maybe prove this by using induction?How do start on a problem like this and where do I need to get to?

In order to prove that it is a Cauchy sequence you have to prove that 
$$|a_{n+p}-a_n|\longrightarrow 0,\text{ when }n\to\infty,p\to\infty,n,p\in\mathbb N$$
In your case , I 'll give you a HINT:
$$|a_{n+p}-a_n|=|a_{n+p}-a_{n+p-1}+a_{n+p-1}-\dots-a_{n+1}+a_{n+1}-a_n|\leq$$
$$|a_{n+p}-a_{n+p-1}|+\dots +|a_{n+1}-a_n|$$
Can you now continue? Express each difference above in terms of $q,a_0,a_1$
