How to Align Equations in the Align and Proof Environment

You have && that shouldn't be there:

\documentclass{article}
\usepackage{amsmath,amsthm}

\begin{document}

\begin{proof}
\begin{gather*}
S_n = \frac{a(1-r^n)}{1-r}\\[2ex]
\begin{align*}
    \lim _{n\to \infty} S_n &= \lim _{n\to \infty} \frac{a(1-r^n)}{1-r}\\
    &= \lim _{n\to \infty} \frac {a}{1-r} - \lim _{n\to \infty} \frac{ar^n}{1-r}\\
    &= \lim _{n\to \infty} \frac {a}{1-r} \tag{since $|r| < 1$}\\
    &= \frac {a}{1-r} \qedhere
\end{align*}
\end{gather*}
\end{proof}

\end{document}

I propose this variant:

\documentclass{article}
\usepackage{mathtools,amsthm}

\begin{document}

\begin{proof}\leavevmode\vspace*{-\dimexpr\baselineskip + \abovedisplayskip + 5pt}
\begin{align*}
S_n & = \frac{a(1-r^n)}{1-r},\\
\shortintertext{so, taking the limits, }
    \lim _{n\to \infty} S_n &= \lim _{n\to \infty} \frac{a(1-r^n)}{1-r}\\
    &= \lim _{n\to \infty} \frac {a}{1-r} - \lim _{n\to \infty} \frac{ar^n}{1-r}\\
    &= \lim _{n\to \infty} \frac {a}{1-r} \qquad(\text{since }|r| < 1)\\
    &= \frac {a}{1-r}. \qedhere
\end{align*}
\end{proof}

\end{document} 

Here's a solution that uses a single aligned environment. This method simplifies the vertical alignment of "Proof" and the first line of the math material.

\documentclass{article}
\usepackage{amsmath,amsthm}
\begin{document}
\begin{proof}\qquad
$\begin{aligned}[t]
\text{Put }S_n &= \frac{a(1-r^n)}{1-r}\,.\\
\text{Thus, }\lim_{n\to\infty} S_n &= \lim_{n\to\infty} \frac{a(1-r^n)}{1-r}\\
    &= \lim_{n\to\infty} \frac {a}{1-r} - \lim_{n\to\infty} \frac{ar^n}{1-r}\\
    &= \lim_{n\to\infty} \frac {a}{1-r} \qquad\text{(since $|r|<1$)}\\
    &= \frac {a}{1-r}\,. &&&& \qedhere
\end{aligned}$
\end{proof}
\end{document}