You can write a proof using {\tt proof} environment. 


The platform will automatically apply the Q.E.D symbol at the end of your proof, and also make a nice bold {\bf Proof} at the beginning of the proof. In case you are curious, Q.E.D. comes from Latin, which means that that was asked to be demonstrated, or \emph{quod erat demonstrandum}. The mathematical symbol looks like this: $\qed$.


You can write a proof using this notation:

\begin{verbatim}
\begin{proof}
 % some typsetting + math goes here
\end{proof}
\end{verbatim}

Here is an example that proves that multiplication in the time domain is the product of a scalar and the convolution in the frequency domain. In other words, prove $f(t)g(t) \ftp \frac{1}{2\pi} (F * G) (\omega)$

\begin{proof}

\begin{align*}
\iftrans{(F*G)(\omega)} &= \frac{1}{2\pi}\int_{-\infty}^{\infty}(F*G)(\omega)e^{i\omega t}d\omega \\
\iftrans{(F*G)(\omega)} &= \frac{1}{2\pi}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}F(W)G(\omega-W) dW e^{i\omega t}d\omega \\
\iftrans{(F*G)(\omega)} &= \frac{1}{2\pi}\int_{-\infty}^{\infty}F(W)\int_{-\infty}^{\infty}G(\omega-W) e^{i\omega t} d\omega dW \\
\iftrans{(F*G)(\omega)} &= \frac{1}{2\pi}\int_{-\infty}^{\infty}F(W)\int_{-\infty}^{\infty}G(\Omega) e^{i(\Omega+W) t} d\Omega dW \\
\iftrans{(F*G)(\omega)} &= \frac{1}{2\pi}\int_{-\infty}^{\infty}F(W)\lr{\int_{-\infty}^{\infty}G(\Omega) e^{i\Omega t} d\Omega }e^{iWt}dW \\
\iftrans{(F*G)(\omega)} &= \frac{1}{2\pi}\int_{-\infty}^{\infty}F(W)e^{iWt}dW \lr{\int_{-\infty}^{\infty}G(\Omega) e^{i\Omega t} d\Omega}\\
\iftrans{(F*G)(\omega)} &= \frac{1}{2\pi}\lr{2\pi f(t)}\lr{2\pi g(t)} \\
\iftrans{(F*G)(\omega)} &= 2\pi f(t)g(t) \\
\iftrans{\frac{1}{2\pi}(F*G)(\omega)} &= f(t)g(t) \\
\end{align*}

Which shows that multiplication in the time domain is the product of a scalar and the convolution in the frequency domain.

\end{proof}
