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I found the following assertion in a book, and I'm having trouble verifying it: $$ a_n\leq C_1 a_0+C_2\sum_{m=0}^{n}a_m $$ implies $$ a_n\leq(C_1+C_2)(1-C_2)^{-n}a_0. $$

All of the $a_i$ are positive, in case it matters. $C_1$ and $C_2$ are also positive, with $C_2<1$.

For $n=0$ or $1$, the implication is trivial. But I don't have any idea on the general case. Anyone have any pointers?

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First off, rearrange the first inequality to $$(1-C_2)a_n\leq(C_1+C_2)a_0+C_2\sum_{m=1}^{n-1}a_m$$ Then, by induction, replace $a_1$ to $a_{n-1}$ with their version of the second inequality.

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