# Asymptotic density and sum of the reciprocals

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