Existence of Euler-Mascheroni Constant/Proof 2

Proof
Consider $\delta : \R_{\ge 1} \to \R$:
 * $\ds \map \delta x := \int _1 ^x \dfrac {\fractpart u} {u^2} \rd u$

where $\fractpart u$ denotes the fractional part of $u$.

Then $\delta$ is increasing, since $\dfrac {\fractpart u} {u^2} \ge 0$.

Furthermore:

Therefore the improper integral:
 * $\ds \alpha := \int _1 ^{+\infty} \dfrac {\fractpart u} {u^2} \rd u = \lim_{x \mathop \to +\infty} \map \delta x$

exists in $\R$.

Thus:

So: