How do i prove the Euler-Mascheroni constant is positive? Define $\gamma=\lim_{n\to\infty} \sum_{k=1}^n 1/k - ln(n)$.
I know that $\gamma$ is nonnegative, but i don't know how to prove that it is positive.

A simple comparison suffices:  consider the functions $$f(x) = \frac{1}{x}, \quad g(x) = \frac{1}{\lfloor x \rfloor}.$$  Clearly, $g(x) \ge f(x)$ for all $x \ge 1$.  Integrating both on $[1, n]$ gives $$\int_{x=1}^n f(x) \, dx \le \int_{x=1}^n g(x) \, dx,$$ or $$\log n \le \sum_{k=1}^{n-1} \frac{1}{k} < \sum_{k=1}^n \frac{1}{k}.$$
