Given the fact that $0 < \alpha < 1$, the sequence $(a_n)$ satisfies $$ \lvert{a_{n+2} - a_{a+1}}\rvert \leq \alpha\lvert{a_{n+1}} - a_{n}\rvert,\quad \forall n \in \mathbb{N}.$$ then $(a_n)$ is a Cauchy sequence and hence convergent.
How can I use this information to show that the sequences $$a_1 = 0,\, a_2 = 1$$ $$a_n = \frac{1}{3} a_{n-1}+\frac{2}{3}a_{n-2}$$ and $$c_1=1, c_{n+1} = \frac{3+2c_n}{3+c_n}$$ are convergent? I have difficulties in proving the sequence satisfy the given inequality, I think use induction is possible but I have no idea how can I start..
hint
For $ (a_n)$.
we have
$$3a_{n+2}=a_{n+1}+2a_n$$
thus
$$3(a_{n+2}-a_{n+1})=-2(a_{n+1}-a_n)$$
So, $$\alpha=\frac 23$$
For the sequence $ (c_n) $, use the canonical form
$$c_{n+1}=2-\frac{3}{3+c_n}$$