We want to prove that harmonic series is divergent.
One elementary solution is an easy consequence of some well known inequality
This is equivalent to
This inequality is also a consequence of Lagrange theorem applied to logarithmic function on interval [n,n+1].
Summing up from 1 to n we get
From the right side of the above relation we have
Now let consider harmonic series
taking limit in both sides we also have
As the partial sum of the harmonic series is a divergent sequence, the harmonic series is divergent.