Let $x_1=0$. $x_{n+1}=ax_{n}+\frac{1}{n}$ where $a>0$. Prove that $x_n$ is bounded iff $0<a<1$.
In the if part, we have $x_{n+1}-x_{n}<\frac{1}{n}$ From this we get $x_n<\sum_{k=1}^{n-1} \frac{1}{k}$. But I cannot show that the sequence is uniformly bounded. No ideas for the only if part.