How to align equations in the equation environment?

Question

I have been looking on how to align equations and I have seen numerous examples with \align and \flalign, however whenever I try to use these in the equation environment, I get an error. I would like to be able to align the equations at the middle equal signs.

Here is an image of the current output:

\begin{proof}
Let   \begin{math} \varepsilon > 0 \end{math} be given and let \begin{math} \delta = \varepsilon \end{math}. It follows that for  \begin{math} I:(X,d^*) \to (R,d) \end{math}

\begin{equation*}
\text{if}\quad\d^*(f,g) = \int_a^b \bigg  |f(t) - g(t)\, \mathrm{d}t \  \bigg  | < \delta 
\end{equation*} 

\begin{equation*}
\text{then}\quad{\bigg  | \int_a^b f(t) - g(t)\, \mathrm{d}t  \bigg  |  =  \bigg  | \lim_{n \to \infty} \sum_{i=1}^{n}{M_i(t_i-t_{i-1})} \bigg  |\leq
\end{equation*} 
\begin{equation*}
\lim_{n \to \infty} \sum_{i=1}^{n}  \bigg  | {M_i(t_i-t_{i-1})} \bigg  |= \int_a^b \bigg  |f(t) - g(t)\, \mathrm{d}t \  \bigg  | < \delta = \varepsilon
\end{equation*} \

\end{proof}

I am now using aligned instead of the equation environment. Here is the code and the error I'm getting:

CODE:

\documentclass{article}
\usepackage{mathtools}
\usepackage{amsthm}
\begin{document}
\theoremstyle{definition}
\newtheorem{Prb1}{Problem}

\begin{Prb1}
 Since the the integral is nothing more that an infinite sum it obeys the triangle inequality. Hence, we may use this property to prove that \begin{math}I:(X,d^*) \to (R,d) \end{math} is continuous. Where \begin{math} d^*(f,g) = \int_a^b f(t) - g(t)\, \mathrm{d}t\end{math} for continuous functions \begin{math}f,g,\end{math} and \begin{math}I(f) = \int_a^b f(t) \, \mathrm{d}t \end{math} \hfill
\end{Prb1}

\begin{proof}

Let   \begin{math} \varepsilon > 0 \end{math} be given and let \begin{math} \delta = \varepsilon \end{math}. It follows that for  \begin{math} I:(X,d^*) \to (R,d) \end{math}

\begin{align*}
\text{if}\quad\d^*(f,g) &= \int_a^b \bigg  |f(t) - g(t)\, \mathrm{d}t \  \bigg  | < \delta \\
\text{then}\quad{\bigg  | \int_a^b f(t) - g(t)\, \mathrm{d}t  \bigg  |  &=  \bigg  | \lim_{n \to \infty} \sum_{i=1}^{n}{M_i(t_i-t_{i-1})} \bigg  |\leq \\
\lim_{n \to \infty} \sum_{i=1}^{n}  \bigg  | {M_i(t_i-t_{i-1})} \bigg  |&= \int_a^b \bigg  |f(t) - g(t)\, \mathrm{d}t \  \bigg  | < \delta = \varepsilon  \end{align*} 
\begin{equation*}
\text{if}\quad\ d^*(f,g) = \int_a^b \bigg  |f(t) - g(t)\, \mathrm{d}t \  \bigg  | < \delta \end{equation*} 

\end{proof}
\end{document}

ERROR:

))))
Runaway argument?
 \text {if}\quad \d ^*(f,g) &= \int _a^b \bigg |f(t) - g(t)\, \mathrm \ETC.
! Paragraph ended before \align* was complete.
<to be read again> 
                   \par 
l.25 

?

Answer 1

There are a number of issues with the code you present. Below is one rewriting that fixes several points followed by some discussion.

\documentclass{amsart}

\begin{document}
\begin{proof}
  Let \begin{math} \varepsilon > 0 \end{math} be given and
  let \begin{math} \delta = \varepsilon \end{math}. It follows that
  for \begin{math} I\colon (X,d^*) \to (R,d) \end{math} if
\begin{equation*}
\mathrm{d}^*(f,g) = \int_a^b \bigl|f(t) - g(t)\bigr|\, \mathrm{d}t <
  \delta
\end{equation*}
then
\begin{multline*}
  \biggl| \int_a^b f(t) - g(t)\, \mathrm{d}t \biggr|
  = \biggl| \lim_{n \to \infty} \sum_{i=1}^{n}{M_i(t_i-t_{i-1})} \bigg| \\
  \leq
  \lim_{n \to \infty} \sum_{i=1}^{n}  \bigl| {M_i(t_i-t_{i-1})}
  \bigr|
  = \int_a^b \bigl|f(t) - g(t)\big|\, \mathrm{d}t < \delta = \varepsilon.
\end{multline*}
\end{proof}

Alternatively we may write the last display as:
\begin{equation*}
  \begin{split}
    \biggl| \int_a^b f(t) - g(t)\, \mathrm{d}t \biggr|
    &= \biggl| \lim_{n \to \infty} \sum_{i=1}^{n}{M_i(t_i-t_{i-1})} \bigg| \\
    &\leq \lim_{n \to \infty} \sum_{i=1}^{n} \bigl| {M_i(t_i-t_{i-1})}
    \bigr| = \int_a^b \bigl|f(t) - g(t)\big|\, \mathrm{d}t < \delta =
    \varepsilon.
  \end{split}
\end{equation*}
\end{document}

As pointed out by the others align is a top level environment that replaces equation. It has a version that can be used inside equation, namely aligned. However, I would not use this in the above situation.

Usually, one would regard the text parts if and then as not part of the equation (although there are circumstances where one might want to include them in the equation for emphasising a special point). I would thus move them into the main text, and then you have a short equation separated by then from a long equation. The long equation can be either set with multline (first line shoved left, second line shoved right), or aligned with the split environment. Both versions are shown above. Note that I am used to splitting lines before relations such as \leq (different countries have some different traditions here).

Now your code contained a \d which was undefined. I guessed this was meant to be a \mathrm d, and changed it accordingly.

For the absolute values, you should tell LaTeX which is a left delimiter and which is a right delimiter, so use \bigl and \bigr and their vairiants instead of just \big (or \left...\right. The placing of these delimiters in relation to the integrals was mathematically incorrect; I have moved them to a position that can make sense. I have also made some of them smaller than you specified.

Finally, for the map I the following colon should be typeset with \colon for correct spacing. Incidentally, this map is not used in the connected sentence.

Answer 2

Environments like align (from the amsmath package) are meant to be used instead of equation, not inside of it.

If you really want to use it inside of an equation, you should use the aligned environment instead.

\begin{align*}
\text{if}\quad d^*(f,g) &= \int_a^b \bigg  |f(t) - g(t)\, \mathrm{d}t \  \bigg  | < \delta  \\
\text{then}\quad\bigg  | \int_a^b f(t) - g(t)\, \mathrm{d}t  \bigg  |  &=  \bigg  | \lim_{n \to \infty} \sum_{i=1}^{n}{M_i(t_i-t_{i-1})} \bigg  |\leq \\
\lim_{n \to \infty} \sum_{i=1}^{n}  \bigg  | {M_i(t_i-t_{i-1})} \bigg  |&= \int_a^b \bigg  |f(t) - g(t)\, \mathrm{d}t \  \bigg  | < \delta = \varepsilon
\end{align*}

Other version (the spacing will be different, and if you need numbering, you should be careful):

\begin{equation*}
\begin{aligned}
\text{if}\quad d^*(f,g) &= \int_a^b \bigg  |f(t) - g(t)\, \mathrm{d}t \  \bigg  | < \delta  \\
\text{then}\quad\bigg  | \int_a^b f(t) - g(t)\, \mathrm{d}t  \bigg  |  &=  \bigg  | \lim_{n \to \infty} \sum_{i=1}^{n}{M_i(t_i-t_{i-1})} \bigg  |\leq \\
\lim_{n \to \infty} \sum_{i=1}^{n}  \bigg  | {M_i(t_i-t_{i-1})} \bigg  |&= \int_a^b \bigg  |f(t) - g(t)\, \mathrm{d}t \  \bigg  | < \delta = \varepsilon
\end{aligned}
\end{equation*}