Showing that a weakly modular function is holomorphic at $\infty$

Question

I am reading "A First Course in Modular Forms" by Diamond and Shurman. In the first chapter I read: "...showing that a weakly modular holomorphic function $f : H \rightarrow C$ is holomorphic at $\infty$ doesn’t require computing its Fourier expansion, only showing that $\lim_{Im(\tau) \rightarrow \infty} f(\tau)$ exists or even just that $f(\tau)$ is bounded as $Im(\tau)\rightarrow \infty$." Can someone explain why showing the latter two is enough for the function to be holomorphic at infinity?