For an introductory analysis class, why is it that many people avoid teaching it using sequences? I am currently taking an honors introduction to analysis course, and it seems to me that to me defining all of the concepts in terms of sequences allows for much cleaner proofs, and the concepts are also far easier to understand. 
Yet when I see most of the analysis questions on here they typically involve people doing epsilon delta proofs that could be significantly shorter using concepts defined in terms of sequences. So I am wondering, why is it that so many people still insist on making their students do epsilon delta proofs all day when there are cleaner and easier alternatives? 
Given that I have not finished my course there might be some instances where these kinds of proofs are more beneficial then others, and if you know of any I would like to hear it. 
 A: I think you're somewhat mistaken. You still have to use $\epsilon-\omega$-style definition for the limit of sequences, so you don't gain that you can avoid that entirely. You will have to prove some important things about sequences using that definition of limit.
Now I agree that some of the results are conveniently proved using a sequence based approach. For other results the sequence approach would in practice lead to a $\epsilon-\delta$-style proof in disguise.
For example proving that $lim_{x\to0}f(x)=0$ would mean to prove that $lim_{n\to\infty}f(x_n)=0$ for all sequences $x_n\to0$ (as $n\to\infty$). Now that in turn would result in a reasoning that given a $\epsilon>0$ we should find a $\omega$ (depending on the sequence) such that $|f(x_n)|<\epsilon$ whenever $n>\omega$, but in finding a such $\omega$ we have to rely on that $|x_n|$ getting sufficiently small (since $x_n$ is an arbitrary sequence and we can't assume anything else about it) and eventually less than the $\delta$ we would use in a normal $\epsilon-\delta$-style proof.
The bottom line I think this is done is because the need of $\epsilon-\delta$ style proof, using that would familiarize the student with that approach (even if some proofs would be easier with sequence based approach). In addition the sequence based approach would probably need some theorems that are not required otherwise - introducing and proving them could outweigh the benefits.
