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Let $a_n$ and $b_n$ be a sequence of non-negative, non-increasing numbers such that $$A(x):=\sum_{n\leq x} a_n \leq \sum_{n\leq x} b_n := B(x)$$ for all $x$.

Can you conclude that $a_n \ll b_n$?

You need the non-negative condition because otherwise you could have $a_n$ going to negative infinity $b_n$ be some something that converges to a positive number.

You need the non-increasing condition because otherwise you can set $b_n=1$ and let $a_n$ be $0$ most places but then large in some places so that there is a subsequence of the $a_n$ that tends to infinity (i.e. $a_{10^n}=n$ but $a_n=0$ for $n$ not a power of $10$).

Are these the only conditions you need to conclude that $a_n\ll b_n$? If not, what other conditions do you need? Any solution or reference would be appreciated.

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    $\begingroup$ What does $\ll$ mean here? $\endgroup$ – symplectomorphic Jul 4 '16 at 8:30
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Take $a_n=b_n$, then you wont have $a_n<b_n$, so probably not $a_n<<b_n$ either

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