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If $\{g_{k}\}_{k\in \mathbb N}$ is a sequence of functions with the properties:

(1) $g_{k}(x)$ is a nonzero continuous and $g_{k }\geq 0$ on $\mathbb R$, for all $k\geq 1$,

(2) the sequence is uniformly bounded on $\mathbb R$ by 1,

(3) $\sum_{n\geq 1} g_{k}(a_{n})\leq C_{k}$, for all $k\geq 1$, where $C_{k}$ is a decreasing sequence converges to 0, and $\{a_{n}\}$ is any real sequence with $a_{n}$ converges to $\infty$.

(4) EDIT: $g_{k}\to 0$ uniformly.

Can we assume without loss of generality that the sequence $g_{k}$ is decreasing, i.e., can we define a new sequence, say $G_{k}$ in terms of $g_{k}$, that satisfies the above (similar) properties?

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I have no idea! – Whitney Jun 9 '12 at 3:32
Well, suppose you have two functions, $p$ and $q$, and you want to replace $q$ by some function that's related to both $p$ and $q$ but never exceeds $p$. Any thoughts on how to do that? – Gerry Myerson Jun 9 '12 at 6:32

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