# Writing a Series of Equalities and Inequalities across Several Lines

What is the convention for writing a series of successive equalities and inequalities across multiple lines? Let me explain. Let $E_k$ denote an expression; for example, $E_0$ could be a sum or an integral. So we write some thing like: $$E_0 \le E_1 = E_2 < E_3 = E_4$$ If $E_k$'s are large expressions, what is the generally accepted way of writing this across several lines?

1. \begin{align*} E_0 &\le E_1 \\ &\le E_2 \\ &< E_3 \\ &< E_4 \end{align*}
2. \begin{align*} E_0 &\le E_1 \\ &= E_2 \\ &< E_3 \\ &= E_4 \end{align*}
3. \begin{align*} E_0 &\le E_1 \\ &\quad = E_2 \\ &< E_3 \\ &\quad = E_4 \end{align*} or something else?
• 2, definitely. 1 means $E_0 \le E_1 \le E_2 < E_3 < E_4$, and 3 is ugly. – TonyK May 13 '15 at 11:04
• It seems ambigious. I read rather "LHS op1 RHS1 and LHS op2 RHS2.. " than "LHS op1 RHS1 op2 RHS2 .." – mvw May 13 '15 at 11:15
• Well, once i had asked the same question to my maths teacher. She said that, when mathematics is done in depth, lots and lots of concepts are to be applied. Which means there should be lots of calculations. Here neat presentation comes into play. If the solution is not neatly presented, it will add stress to a person and create mistake. So, while solving a lengthy question make sure that there is atmost one inequality or equality symbol in one line. – Aditya Kumar May 13 '15 at 12:24
• I agree with @TonyK. 2 is the best and 3 is ugly. 1 is probably wrong since it claims that $E_{3}<E_{4}$ and you said $E_{3}=E_{4}$. The only thing I'd add is that $E_{0}\leq E_{1}=E_{2}<E_{3}=E_{4}$ is also just fine provided that the expressions are not so long that they bleed into the margins. The one-line string of inequalities is nice if they all fit in a single line. I usually only split it up when there are enough terms (or complicated enough terms) that they do not fit in one line. – TravisJ May 13 '15 at 12:28
• One other aside, if you're writing a paper to be published, 2 leaves a lot of white space on the paper (unless the expression $E_{i}$ are long). If this adds un-necessary length to the paper, a publisher may not appreciate it. This is particularly true in CS (where I currently work) when paper submissions all have very strict length requirements (must be fewer than X pages). – TravisJ May 13 '15 at 12:31