How to sequentially align equations around differing mathematical relations in LaTeX

Philip Thomas K.

17 June 22


I feel like there probably is a better, more succinct way to convey what exactly I want. If I ever do find such a way, I might change the title of this post, because the way it stands, it just sounds like the title of some esoteric article on a math formatting forum.

Anyroads, I came across this proof:

A proof of the algebraic limit theorem for sequences

Which is alright, but I strongly believed it could be formatted in a much nicer way, like this:

A proof of the algebraic limit theorem for sequences, done right.

It’s far more succinct, a bit more rigorous, and in my eyes, formatted correctly. But to get it into this manner was not something that was obvious to me.

I had no idea what to type into any search engine to explain what I wanted. I did type something like “sequentially aligned equations latex”, but to no avail. That’s exactly why I had to make this post, because even my online search skills could not yield anything satisfactory.

By sheer luck, I thought up this simple solution

\forall\varepsilon>0,\ \exists N\in\mathbb{N}\backepsilon n>N & \implies |a_n -a|, |b_n-b| < \frac{\varepsilon}{2}\\
& \begin{aligned}
    {} \implies |a_n +b_n -(a+b)| & \leqslant|a_n -a|+|b_n-b|\\
                      & < \frac{\varepsilon}{2} + \frac{\varepsilon}{2}\\
                      & = \varepsilon

Which yields:

\[\begin{align} \forall\varepsilon>0,\ \exists N\in\mathbb{N}\backepsilon n>N & \implies |a_n -a|, |b_n-b| < \frac{\varepsilon}{2}\\ & \begin{aligned} \implies |a_n +b_n -(a+b)| & \leqslant|a_n -a|+|b_n-b|\\ & < \frac{\varepsilon}{2} + \frac{\varepsilon}{2}\\ & = \varepsilon \end{aligned} \end{align}\]

As required. The reason this works, is because adding {} before a mathematical relation sorts the spacing out nicely. Theoretically, you could keep going, but then your equations will probably get pretty wide pretty quickly, for now, two sequential re-alignments seem to work out just nice.

Also, yes, I use the “backepsilon” to mean “such that”, it just seems so concise, but no, I do not write it.

Best Regards,