Euler's Integral Theorem/Proof 2

Proof
Recall the :

Therefore:
 * $0 \le x - \floor x < 1$

For all $n \in \N_{>0}$:

Therefore:
 * $\ds \forall n \in \N_{>0} : 0 \le \size {H_n - \ln n - \gamma} \lt \dfrac 1 n$