We'll prove (approximately) that log(n) is big-O of Q_n as follows.
For each n , let H_n be the n-th Harmonic number,
i.e. the sum of 1/k for k from 1 to n.
Now prove by induction that Q_n is bigger than (1/2) * H_n.
Deduce that Q_n goes to infinity.
Then explain where your proof would break down if you tried to prove that S contained all integers greater than 20.