In Diamond and Shurman's book A First Course in Modular Forms, exercise 5.1.4 asks you to show that if $g_1,\ldots, g_d:\mathcal{H}\to \mathbb{C}$ are holomorphic at infinity, then so is is their sum $g_1+\cdots +g_d$.
The hint in the book is to let the period be $h=\text{lcm}(\{h_j\})$ where $h_j$ is the period for $g_j$. Now using this it's clear to me that indeed by summing up all Fourier expansions of the $g_j$'s we get a valid expansion in $q=e^{2\pi i \tau/h}$.
However, I was wondering if the following alternative reasoning works. In the beginning of the book the authors say that to be holomorphic at $\infty$ it's actually equivalent to be bounded as $\text{Im}(\tau)\to \infty$. So could we not just say that $$|g_1+\cdots +g_d|\leq |g_1|+\cdots +|g_d|$$ and since each of the $g_j$ are bounded as $\text{Im}(\tau)\to \infty$, the same holds for their sum?
Thank you!