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Let $\{c_k\}_{k=0}^{\infty}$ a sequence of positive numbers such that there exists $K>0$ with $c_m=0$ for all $m\geq K$ and $$c_k \leq c_{k-1}+2a^{k}c_{2k} \mbox{ for all } k\geq 1$$ with $\frac{1}{2}<a<1$ a constant.

Then $c_k \leq Ac_0$ for all $k\geq 0$ for some fixed constant $A>0$.

I tried the substitution $d_k=\frac{c_k}{k+1}$, to have $d_k < d_{k-1}\frac{k}{k+1} + d_{2k}\frac{1}{k+1}$ if $4a^{k} < \frac{1}{k+1}$, but this happens for large $k$. I don't know what to do for small $k$. I would appreciate any help.

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