Let $\{a_n\}_n^{\infty}, \{b_n\}_n^{\infty}$ be sequences of nonnegative numbers and $0 \leq q < 1$, so that $ a_{n+1} \leq qa_n + b_n, \quad\text{for all}\quad n \geq 0. $ Prove that
(i) If $\displaystyle\lim_{n\rightarrow\infty} b_n = 0$, then $\displaystyle\lim_{n\rightarrow\infty}a_n = 0.$
(ii) If $\displaystyle\sum_{n=0}^{\infty}b_n<\infty,$ then $\displaystyle\sum_{n=0}^{\infty}a_n<\infty$.