I refer to the following lecture notes on theta functions: https://web.math.princeton.edu/~gunning/theta2/A
On page 6 Section 2 of these notes (page 13 in the pdf), the definition of a theta function is given and in Lemma 3, it is proven that theta functions converge uniformly on compact subsets. Can anyone please explain why the following estimate used in the lemma holds?
$ \exp (-2\pi \left( \frac{1}{2} n^T Y n + n^T y \right)) \leq \exp (-2\pi \left( \frac{1}{2} bn^T n + an \right) ) $