Existence of Euler-Mascheroni Constant/Proof 2

Proof
By Integral Expression of Harmonic Number:
 * $\ds \sum_{k \mathop = 1}^n \frac 1 k = 1 + \int _1 ^n \dfrac {\floor u} {u^2} \rd u$

Therefore:

In addition, $\ds \sequence {\sum_{k \mathop = 1}^n \frac 1 k - \ln n}$ is a decreasing sequence, since $u - \floor u \ge 0$ in $\paren 1$.

Therefore by monotone convergence theorem, this sequence converges to a limit in $\R_{\ge 0}$.