On Dave Mumford's Tate Lectures on Theta I, he begins by proving that $\theta(z,\tau)$ converges. It begins something like:
Let $|Im(z)|<c$ and $Im(\tau)>\epsilon$, then:
$|e^{\pi i n^2 \tau + 2\pi inz}| < (e^{-\pi \epsilon })^{n^2} (e^{2\pi c})^{n}$
It is clear to me that $\displaystyle |e^{\pi i n^2 \tau + 2\pi inz}|=(e^{-\pi Im(\tau) })^{n^2} (e^{-2\pi Im(z)})^{n}$.
I understand the rest of the proof, assuming the inequality, but I don't get how he get's it in the first place. Thanks in advance for any help!
I'm not really sure what your issue is. Maybe this:
The function $R\ni x\mapsto e^{-x}$ is decreasing. Thus, since $\text{Im}\,\tau>\epsilon$ it holds that $$ e^{-\pi\text{Im}\,\tau}<e^{-\pi\epsilon}. $$ Also, since $\text{Im}\,z>-c$ $$ e^{-2\pi\text{Im}\,z}<e^{-2\pi(-c)}=e^{2\pi c}. $$