Showing a recursive sequence is Cauchy 5

Question

$(b_n)$ is a sequence such that $b_1 = 1$ and $b_{n+1} = − \dfrac{1}{5}(b_n−2)$ for all $n \in \mathbb{N}$. Show that $(b_n)$ is a Cauchy sequence.

I can see that the sequence converges to $\dfrac{1}{3}$ but I am struggling to prove this using the definition of a Cauchy sequence.

Answer 1

Hint

observe that

$$b_{n+1}-\frac{1}{3}=-\frac{1}{5}\left(b_n-\frac{1}{3}\right)$$

$$=\frac{2}{3}\frac{(-1)^n}{5^n}$$