Existence of Euler-Mascheroni Constant

Theorem
The sequence:
 * $\displaystyle \left \langle \sum_{k \mathop = 1}^n \frac 1 k - \ln n \right \rangle$

converges to a limit.

This limit is known as the Euler-Mascheroni constant.

Proof
Let $f:\R\setminus\left\{0\right\} \to \R, \displaystyle f \left({x}\right) = \frac 1 x$.

Clearly $f$ is continuous and positive on $\left[{1 \,.\,.\,+\infty}\right)$.

From Reciprocal Sequence Strictly Decreasing, $f$ is decreasing on $\left[{1 \,.\,.\, +\infty}\right)$.

Therefore the conditions of the Euler-Maclaurin Summation Formula hold.

Thus the sequence $\left \langle {\Delta_n} \right \rangle$ defined as:
 * $\displaystyle \Delta_n = \sum_{k \mathop = 1}^n f \left({k}\right) - \int_1^n f \left({x}\right) \ \mathrm d x$

is decreasing and bounded below by zero.

But from the definition of the natural logarithm (or from Equivalence of Logarithm Definitions):
 * $\displaystyle \int_1^n \frac {\mathrm d x} x = \ln n$

Hence the result.