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\newtheorem{thm}{Theorem}

\begin{document}

\begin{thm}
If $v$ and $w$ are vectors in $\mathbb{R}^{n}$ equipped with the standard inner product, then
\begin{equation} \label{eq:CS}
 \left| \langle v , w \rangle \right| \leq |v| |w|
\end{equation}
\end{thm}
\begin{proof}
Writing $v = (v_1, \ldots, v_n)$ and $w = (w_1, \ldots, w_n)$, the inner product is given by the sum
\begin{equation}
   \langle v , w \rangle  = \sum_{i=1}^{n} v_i w_i,
\end{equation}
while the norm of $v$ and $w$ are respectively given by 
\begin{align}
  |v| & = \sqrt{ \langle v , v \rangle   } \\
 |w| & = \sqrt{ \langle w, w \rangle  }.
\end{align}
We shall prove the Cauchy-Schwarz inequality \eqref{eq:CS} by proving the corresponding inequality for the squares.  We start with the square of the right hand side, which, by the above formula for the norm, is equal to
\begin{equation} \label{eq:1}
  \left( \sum_{i=1}^{n}v_{i}^{2} \right)  \left( \sum_{j=1}^{n}w_{j}^{2} \right) = \sum_{i=1}^{n} v_{i}^{2} w_{i}^{2} +   \sum_{i=1}^{n}  \sum_{j=1}^{i-1} \left( v_{i}^{2} w_{j}^{2} + v_j^{2} w_{i}^{2} \right).
\end{equation}
On the other hand, the square of the left hand side is
\begin{equation} \label{eq:2}
  \left( \sum_{i=1}^{n}v_{i} w_{i} \right)^{2} = \sum_{i=1}^{n} v_{i}^{2} w_{i}^{2} +  \sum_{i=1}^{n}  \sum_{j=1}^{i-1} 2 v_{i} w_{i}v_{j} w_{j}.
\end{equation}
The difference between Equations \eqref{eq:1} and \eqref{eq:2} is equal to
\begin{equation}
   \sum_{i=1}^{n}  \sum_{j=1}^{i-1} \left( v_{i}^{2} w_{j}^{2} -   2 v_{i} w_{i}v_{j} w_{j}+ v_j^{2} w_{i}^{2}  \right)  =  \sum_{i=1}^{n}  \sum_{j=1}^{i-1} (v_i w_j  - v_j w_i)^2.
\end{equation}
Since this expression is a sum of squares, it is non-negative.  In particular, the left hand side of Equation \eqref{eq:1} is greater than  or equal to that of Equation \eqref{eq:2}
\begin{equation}
    \left( \sum_{i=1}^{n}v_{i}^{2} \right)  \left( \sum_{j=1}^{n}w_{j}^{2} \right)  \geq \left( \sum_{i=1}^{n}v_{i} w_{i} \right)^{2}.
\end{equation}
Taking the square root of each side, we conclude, as desired, the Cauchy-Schwarz inequality appearing in Equation \eqref{eq:CS}.
\end{proof}
\end {document}