\documentclass{article}
\usepackage{amsmath,amssymb}
\newcommand{\justif}[2]{&{#1}&\text{#2}}
\newcommand{\conv}{\ast}

\begin{document}
\section*{Question 1}

\subsection{Show that for any discrete-time signal $v$, $(v \conv \delta)[n] = v[n]$ }
  Proof:
  \begin{alignat*}{2}
  \text{Let} f[n] &= (v \conv \delta)[n] \\
  f[n] &=\sum_{i = -\infty}^{\infty} v[i] \delta[n-i]  &\quad &\text{by definition of convolution}\\
       &= v[n]                                         &\quad &\text{by sifting theorem}\\
  \end{alignat*}


\subsection{Show that for any discrete-time signal $v$, $(v \conv Delay_M(\delta))[n] = Delay_M(v)$ }
  Proof:
  \begin{alignat*}{2}
  &\text{Let $v[n]$ be a discrete-time signal} \\
  &\text{Let $f[n] = (v \conv Delay_M(\delta))[n]$} \\
  f[n] &=\sum_{i = -\infty}^{\infty} v[i] Delay_i(Delay_M(\delta))  &\quad &\text{by definition of convolution}\\
  f[n] &=\sum_{i = -\infty}^{\infty} v[i] \delta[n-M-i ] \\
  &\text{We introduce a change of varaible, $o = n - M$} \\
  f[n] &=\sum_{i = -\infty}^{\infty} v[i] \delta[o-i ] \\
       &= v[o]                                         &\quad &\text{by sifting theorem}\\
       &= v[n-M] \\
       &= Delay_M(v)  &\quad &\text{by Definition of $Delay_M$}\\
  \end{alignat*}

\subsection{Show that for any two discrete-time signals $v$ and $w$, and any integer $M$, $v \conv Delay_M(w) = Delay_M(v \conv w) $}
  Proof:
  \begin{alignat*}{2}
  &\text{Let $v[n]$ and $w[n]$ be discrete-time signals} \\
  &\text{Let $f[n] = (v \conv Delay_M(w))[n]$} \\
  f[n] &=\sum_{i = -\infty}^{\infty} v[i] Delay_i(Delay_M(w))  &\quad &\text{by definition of convolution}\\
  f[n] &=\sum_{i = -\infty}^{\infty} v[i] w[n-M-i ] \\
  &\text{We introduce a change of varaible, $o = n - M$} \\
  f[n] &=\sum_{i = -\infty}^{\infty} v[i] w[o-i ] \\
       &= (v \conv w)[o]                                         &\quad &\text{by definition of convolution}\\
       &= (v \conv w)[n-M] \\
       &= Delay_M(v \conv w)  &\quad &\text{by Definition of $Delay_M$}\\
  \end{alignat*}

\subsection{Show that for any discrete-time signal $v$, $Flip(Delay_M(v)) = Delay_{-M}(Flip(v))$}
  \begin{alignat*}{2}
  \text{Let} f[n] &= Flip(Delay_M(v)) \\
                  &= Flip(v[n-M])        &\quad &\text{by Definition of $Delay_M$}\\
                  &= v[-(n-M)]           &\quad &\text{by Definition of $Flip$}\\
                  &= v[-n+M] \\
                  &= Delay_{-M}(v[-n])   &\quad &\text{by Definition of $Delay_M$}\\
                  &= Delay_{-M}(Flip(v)) &\quad &\text{by Definition of $Flip$}\\
  \end{alignat*}

\section*{Question 2}
\end{document}