Existence of Euler-Mascheroni Constant

Theorem
The sequence $$\left \langle \sum_{k=1}^n \frac 1 k - \ln n \right \rangle$$ converges to a limit.

This limit is known as the Euler-Mascheroni constant.

Proof
Let $$f \left({x}\right) = \frac 1 x$$.

Clearly $$f$$ is continuous, positive and 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 $$\Delta_n = \sum_{k=1}^n f \left({k}\right) - \int_1^n f \left({x}\right) dx$$ is decreasing and bounded below by zero.

But from the definition of the natural logarithm, $$\int_1^n \frac {dx} x = \ln n$$.

Hence the result.