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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 $\displaystyle\sum_{n=0}^{\infty}a_n<\infty$.

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    $\begingroup$ This question (1) gives orders and (2) does not indicate what the OP tried and where the OP is stuck. The other way round would be preferable. $\endgroup$ – Did Mar 30 '12 at 8:10
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Hints:

(i) Try to find an equivalent series for $a_n$, note that $b_n \to 0$ implies that $b_n$ is bounded (why?), and combine these results to find that $a_n$ is dominated by a geometric series.

(ii) Use the expansion from (i) to derive that some multiple of $\sum_n b_n$ dominates $\sum_n a_n$.

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