Existence of Euler-Mascheroni Constant/Proof 2

Proof
For $n \in \N_{>0}$ let:
 * $\ds \gamma_n := \sum_{k \mathop = 1}^n \frac 1 k - \ln n$

Then:

On the other hand:

Thus by monotone convergence theorem, the sequence $\sequence {\gamma_n}$ converges to a limit in $\R_{\ge 0}$.