In lecture, my professor stated that $ \lim_{N \to \infty} \int_{\gamma_N} f(z) \cot(\pi z) \; dz = 0 $
where $\gamma_N$ is the square contour with vertices at $\pm (N + \frac12) \pm i(N + \frac12)$ and $f$ is complex-valued function (not necessarily entire) such that $|z f(z)| < M$ for sufficiently large $|z|$.
I can't see how to prove this. If $|z^2f(z)|$ is bounded, then you can bound the cotangent on the contour and get the desired limit, but I don't know how to manage it with the weaker constraint given. Any advice would be appreciated. Thanks.