How to correctly format (and align) a LaTeX proof?

Question

I'm new to LaTeX and I'm trying to figure out how to correctly format a proof. Can someone show me a basic template for how to do this?

The part I am having the most trouble with is creating new columns and customizing alignment... for instance, how do I add a right-aligned annotation (i.e. by distributive property) to a line of this sample proof? Should I use eqnarray* or is there a better option?

\begin{proof}
  Let $t,u \in \mathbb{R}$ where $t=xy$ and $u=zw$. So,
  \begin{eqnarray*}
    4xyzw &=& 2\cdot2tu \\
    &\le& 2\cdot(t^2+u^2) \\
    &=& 2\cdot((xy)^2+(zw)^2) \\
    &=& 2\cdot(x^2y^2+z^2w^2) \\
    &=& 2x^2y^2+2z^2w^2 \\
    &\le& ((x^2)^2+(y^2)^2)+((z^2)^2)+(w^2)^2) \\
    &=& x^4+y^4+z^4+w^4
  \end{eqnarray*}
\end{proof}

Answer 1

\documentclass{article}

\usepackage{amsmath,amsthm,amssymb}

\begin{document}

\begin{proof}
  Let $t,u \in \mathbb{R}$ where $t=xy$ and $u=zw$. So,
  \begin{align*}
    4xyzw &= 2\cdot2tu \\
    &\le 2\cdot(t^2+u^2) \\
    &= 2\cdot((xy)^2+(zw)^2) &&\text{(substituting variables)} \\
    &= 2\cdot(x^2y^2+z^2w^2) \\ 
    &= 2x^2y^2+2z^2w^2 \\
    &\le ((x^2)^2+(y^2)^2)+((z^2)^2)+(w^2)^2) \\
    &= x^4+y^4+z^4+w^4 &&\qedhere
  \end{align*}
\end{proof}

\end{document}

The alignment is better (eqnarray should never be used for serious mathematical writing) and, moreover, the "end-of-proof" can be placed aligned with the last equation; \qedhere is necessary only when the proof ends with an alignment environment or with a list (enumerate, itemize or description); the && before \qedhere is only necessary when there are other comments.

I didn't check the math, though. ;-)

Answer 2

Adding to egreg's remarks: to add right-aligned annotation — in parentheses — you can use the \tag macro; to have annotation without parentheses, use the \tag* macro. (This must be given before the line-break; for this reason, I like to put the line break immediately before the following line, but that's just my personal style.) And again, as egreg noted, you should use \qedhere on the last line of an equation environment, if your proof ends at an equation.

\begin{align*}
  4xyzw 
  &= 2\cdot2tu 
  \\ &\le 2\cdot(t^2+u^2)                    \tag{a remark in parentheses}
  \\ &= 2\cdot((xy)^2+(zw)^2)
  \\ &= 2\cdot(x^2y^2+z^2w^2)                \tag*{a remark without parentheses}
  \\ &= 2x^2y^2+2z^2w^2
  \\ &\le ((x^2)^2+(y^2)^2)+((z^2)^2)+(w^2)^2)
  \\ &= x^4+y^4+z^4+w^4                      \qedhere
\end{align*}

If you find yourself having multi-line equations where you want equation numbers — but only on the last line — the macro \notag will also come in handy. (I insert them just before the line-breaks, to make it easy to copy the sequence \notag \\ &= for multi-line equations.)

\begin{align}
  4xyzw 
  &= 2\cdot2tu 
  \notag\\ &\le 2\cdot(t^2+u^2)                    \tag{a remark in parentheses}
  \\ &= 2\cdot((xy)^2+(zw)^2)
  \notag\\ &= 2\cdot(x^2y^2+z^2w^2)                \tag*{a remark without parentheses}
  \\ &= 2x^2y^2+2z^2w^2
  \notag \\ &\le ((x^2)^2+(y^2)^2)+((z^2)^2)+(w^2)^2)
  \notag \\ &= x^4+y^4+z^4+w^4
\end{align}

Edited to add: don't use \tag or \tag* for remarks, unless you're happy with them appearing on the left (instead of on the right) whilst using the lefteqn option.

Answer 3

You can also use the package witharrows specially written in this aim.

\documentclass{article}
\usepackage{amsthm,amssymb}
\usepackage{witharrows}
\begin{document}
\begin{proof}
  Let $t,u \in \mathbb{R}$ where $t=xy$ and $u=zw$. So,
  \begin{DispWithArrows*}
    4xyzw &= 2\cdot2tu \\
    &\le 2\cdot(t^2+u^2) \Arrow[i]{substituting variables}\\
    &= 2\cdot((xy)^2+(zw)^2)  \\
    &= 2\cdot(x^2y^2+z^2w^2) \\ 
    &= 2x^2y^2+2z^2w^2 \\
    &\le ((x^2)^2+(y^2)^2)+((z^2)^C2)+(w^2)^2) \\
    &= x^4+y^4+z^4+w^4 \qedhere
  \end{DispWithArrows*}
\end{proof}
\end{document}

Answer 4

One may also use the alignat environment. See the following:

\documentclass[letterpaper]{article}
\usepackage{amsmath,amssymb}
\newcommand{\justif}[2]{&{#1}&\text{#2}}
\begin{document}
\begin{alignat*}{2}
4xyzw &= 2\cdot2tu                              \justif{\quad}{remark}\\
      &\le 2\cdot(t^2+u^2)                      \justif{\quad}{remark}\\
      &= 2\cdot((xy)^2+(zw)^2)                  \justif{\quad}{remark}\\
      &= 2\cdot(x^2y^2+z^2w^2)                  \justif{\quad}{remark}\\
      &= 2x^2y^2+2z^2w^2                            \\ % no remark
      &\le ((x^2)^2+(y^2)^2)+((z^2)^2)+(w^2)^2)     \justif{\quad}{remark}\\
      &= x^4+y^4+z^4+w^4          
\end{alignat*}
\end{document}

Note the introduction of a \justif{<horizontal space>}{<content>} command that takes as arguments the separation and content (remark).

Answer 5

This is not an elegant solution but might be an option nevertheless. You can also fake it with a tabular. This way, you can print some borders if you want (just in case of those provide-the-missing-part tests so loved by some high school math teachers).

\documentclass[10pt]{article}

\usepackage[margin=1in]{geometry}
\usepackage{amsmath}
\usepackage{array}
\newcolumntype{L}{@{\hphantom{.}}>{$}l<{$}}
\renewcommand{\arraystretch}{1.2}
\begin{document}

\begin{tabular}{LLll}
\lvert p-q\rvert & =\sqrt{(p-1)^2} & \hphantom{some text} & by definition of square root\\
    &=\sqrt{p^2 -2pq +q^2} & &  by multiplication\\
    &=\sqrt{p^2-2pq+q^2 +2pq -2pq} & & by the additive identity\\
    &=\sqrt{p^2+2pq+q^2 -4pq} & &     by grouping like terms\\
    &=\sqrt{(p+q)^2 -4n} & &          by the distributive law\\
\end{tabular}

\end{document}