Prove that a sequence give by a recurrence relation converges

Question

let $b_n$ be a real sequence which satisfies

$b_{n+1}=(\frac{1}{n^2}-\frac{1}{2})b_n -\frac{1}{3n^2}$, $b_1=\frac{2}{3}$

I want to show that $b_n$ converges to $0$, which seems to be true according to my simulation.

I have tried to use some general ways to proof the convergence of a sequence like ratio test, but could not finish.

Answer 1

Note that for all $\varepsilon>0$, we have $|b_{n+1}|\leq\frac 23|b_n|+\varepsilon$ eventually, hence $$\limsup |b_n|\leq 3\varepsilon$$