Does there exist a proof, not by induction, of the fact that $1 + (1/2) + \cdots + (1/n)$ is not an integer for any $n > 1$? In all the books I have read the proof is done by induction and is not quite illuminating . Without induction the most I have been able to get is ;
$(d(1) + d(2) + \cdots + d(n) ) /n ≤ 1 + (1/2) + \cdots + (1/n) < 1+ (d(1) + d(2) + \cdots + d(n) )/n$; where $d(k)$ denotes the number of positive divisors of $k$ , can this inequality be improved or, adequately altered to prove the theorem ? Does there exist a proof of the theorem which does not use Bertrand's postulate neither order properties nor induction ?