Proving the harmonic series, multiplied by a factor of 1/n, decreases monotically to zero. I would like to show that $\left(1 + \frac 12 + \dots + \frac 1n\right) \cdot \left(\frac 1n\right)$ decreases monotonically to zero.
I have seen one method: to first show that the difference $\left(1 + \frac 12 + \dots + \frac 1n\right) - \log(n)$ decreases monotonically to the Euler-Mascheroni constant, but I was wondering if there is another (perhaps cleaner or more instructive) way of proving the limit is zero.
Any help would be greatly appreciated.
Thanks,

This is the arithmetic mean of the numbers $1, 1/2,\ldots,1/n$. If you add a number that is lesser than the others, say $1/(n+1)$, the mean decreases.
The limit is $0$ because
$$\frac{H_n}n<\frac{\log n}n\to0$$
