Equation number within proof environment while aligning all the equations?

Question

I know equation numbering is possible in a proof environment, as shown below in MWE1:

\documentclass[12pt,a4paper]{article}
\usepackage[latin1]{inputenc}
\usepackage{amsmath}
\usepackage{amsthm}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage[left=2.00cm, right=2.00cm, top=2.00cm, bottom=2.00cm]{geometry}
\begin{document}
\begin{proof}
A+B=C
\begin{equation}
G(t)=L\gamma!\,t^{-\gamma}+t^{-\delta}\eta(t) \qedhere
\end{equation}
\end{proof}
\end{document}

But I want the equations to be aligned. Thus I need \begin{align}...\end{align}, as shown in the MWE2 below:

\documentclass[12pt,a4paper]{article}
\usepackage[latin1]{inputenc}
\usepackage{amsmath}
\usepackage{amsthm}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage[left=2.00cm, right=2.00cm, top=2.00cm, bottom=2.00cm]{geometry}
\begin{document}
\begin{proof}
\begin{align}
\mu &= np \nonumber\\
&= \textonehalf n;\nonumber\\
\begin{equation}
\sigma &= \sqrt{np\left(1-p\right)}\\
\end{equation}
&=  \textonehalf \sqrt{n};\nonumber\\
Z &= \dfrac{X-\mu}{\sigma} \nonumber\\
&= \frac{X-\textonehalf n}{\textonehalf \sqrt{n}};\nonumber\\
\vartriangle Z&= \frac{\left(X+1\right)-\textonehalf n}{\textonehalf 
\sqrt{n}} - \frac{X-\textonehalf n}{\textonehalf \sqrt{n}} \nonumber\\
\begin{equation}
\vartriangle Z&= \dfrac{1}{\textonehalf \sqrt{n}}\\
\end{equation}
\lim_{n \to \infty} \vartriangle Z &= 0
\end{align} 
\end{proof}
\end{document}

But there is an error in MWE2. How do I assign equation numbers to some selected equations in a proof environment while aligning all the equations? I want the equation number so that I can label (\label{}) such equations and reference (\ref{}) it within the document. I want the alignment along the equation signs, so that it will look aesthetical and professional.

Answer 1

As Zarko mentioned, the equation environment must not be used in an align environment. Just removing the equation environment works for me:

\documentclass[12pt,a4paper]{article}
\usepackage[latin1]{inputenc}
\usepackage{amsmath}
\usepackage{amsthm}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage[left=2.00cm, right=2.00cm, top=2.00cm, bottom=2.00cm]{geometry}

\newcommand{\numberthis}{\refstepcounter{equation}\tag{\arabic{equation}}}
\newcommand{\labeln}[1]{\numberthis\label{#1}}
\newcommand{\mathonehalf}{\ensuremath{\frac{1}{2}}}

\begin{document}
\begin{proof}
\begin{align}
    \mu &= np \\
    &= \mathonehalf n;\\
    \sigma &= \sqrt{np\left(1-p\right)} \labeln{eq:sigma}\\
    &=  \mathonehalf \sqrt{n};\\
    Z &= \dfrac{X-\mu}{\sigma}\\
    &= \frac{X-\mathonehalf n}{\mathonehalf \sqrt{n}};\\
    \vartriangle Z&= \frac{\left(X+1\right)-\mathonehalf n}{\mathonehalf 
    \sqrt{n}} - \frac{X-\mathonehalf n}{\mathonehalf \sqrt{n}}\\
    \vartriangle Z&= \dfrac{1}{\mathonehalf \sqrt{n}} \labeln{eq:dz}\\
    \lim_{n \to \infty} \vartriangle Z &= 0 \labeln{eq:limdz}
\end{align} 
\end{proof}

References to $\sigma$ \eqref{eq:sigma} and $\Delta Z$ \eqref{eq:dz}.
\end{document}

Answer 2

Is it this you want? You can get small fractions in math mod ewith \tfrac. But in this context, I'd recommend the \mfrac command from  nccmath.

Also: preferably, use the utf8 encoding nowadays. Don't nest an equation environment in align. Needless to load amsfonts if you load amssymb. Just use this simplified code:

\documentclass[12pt, a4paper]{article}
 \usepackage[utf8]{inputenc}
 \usepackage{textcomp}
 \usepackage{amsmath, nccmath}
 \usepackage{amsthm}
 \usepackage{amssymb}
 \usepackage[margin=2cm]{geometry}

 \begin{document}

 \begin{proof}
 \begin{align}
 \mu &= np \nonumber\\
 &= \tfrac{1}{2}n;\nonumber\\
 \sigma &= \sqrt{np (1-p )}\\
 &= \mfrac{1}{2} \sqrt{n};\nonumber\\
 Z &= \dfrac{X-\mu}{\sigma} \nonumber\\
 &= \frac{X-\textonehalf n}{\textonehalf \sqrt{n}};\nonumber\\
 \Delta Z&= \frac{\left(X+1\right)-\textonehalf n}{\textonehalf
\sqrt{n}} - \frac{X-\textonehalf n}{\textonehalf \sqrt{n}} \nonumber\\
\Delta Z&= \dfrac{1}{\textonehalf \sqrt{n}}\\
\lim_{n \to \infty} \Delta Z &= 0
\end{align}
\end{proof}

\end{document}