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Let A and B be two infinite proper-subsets of the set of positive integers. Let A(n) denote the number of those elements of the set A , which does not exceed n ; we use similar definition for B(n) . Also let lim A(n)/n > lim B(n)/n , as n→∞

If the sum of the reciprocals of the numbers in B is divergent then can we ever conclude that the sum of the reciprocals of the numbers in A is also divergent ?

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up vote 2 down vote accepted

Your inequality implies $\lim A(n)/n$ exists and is positive, which is enough to conclude divergence for $A$, regardless of what happens to $B$.

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So, if lim A(n)/n exists and is > 0 then I can always conclude that the sum of the reciprocals is divergent? –  Souvik Dey Oct 11 '12 at 6:04
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@Souvik SeeTheorem on natural density. –  Martin Sleziak Oct 11 '12 at 6:18
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