Proving a Functional Equation for the JacobiTheta Function

Question

Let $\Theta(t) = \sum_{k = - \infty}^{\infty} e^{-\pi k^{2} t} $. How can it be proved that $\Theta(\frac{1}{t}) = \sqrt{t}\Theta(t)$? I have read a proof here https://scholarship.claremont.edu/cgi/viewcontent.cgi?article=1192&context=hmc_theses (pp.36-42), but the arguments used to prove lemmas 4.2.3 and 4.2.4 did incorrectly assumed that if $a_{k, l} \rightarrow a_{k}$ for fixed $k$ as $l \rightarrow \infty$, then $\sum_{k = -l}^{l} a_{k, l} \rightarrow \sum_{k = -\infty}^{\infty} a_{k}$. Is there another way of proving this functional equation for Jacobi's Theta Function?