Let with n>1. Prove that 1 + 1/2 + 1/3 + .....1/n
Let be the greatest prime not exceeding .
So we must have: (1) . Suppose this were false, then but, by Bertrand's Postulate there's a prime such that which is absurd since is the greatest prime not exceeding .
Now suppose then since is an integer
By (1) we have that doesn't divide so is not an integer. Absurd
There is a classical proof provided by Nicole Oresme, where you consider the subsequence :
And therefore, in general:
.
And since the subsequence is unbounded, the sequence diverges.
And therefore it is sufficient to say that if the sequence diverges, no integer can be reached and therefore: