Euler's Integral Theorem/Proof 2

Proof
Recall the definition of the floor function:

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

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

From Existence of Euler-Mascheroni Constant Proof 1, we have:
 * $\ds \Delta_n = \sum_{k \mathop = 1}^n \dfrac 1 k - \int_1^n \dfrac 1 x \rd x$

is decreasing and bounded below by zero.

Therefore:
 * $H_n - \ln n \ge 0$

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