Right, other U. call the course Elementary Analysis and the content might be different.
A couple of books for the entirely uninitiated would be:
1) A First Course in Mathematical Analysis, by Burkill, pub. Cambridge, and
2) Yet Another Introduction to Analysis, by Bryant, pub. Cambridge
I think Bryant is used at Baruch. Burkill used to be recommended reading for incoming math students at Cambridge (i.e., before they started). Neither even defines Cauchy sequences. I prefer Burkill myself because it's a nifty little book and explains how analysis can elucidate the properties of the exponential and trig functions, and how completeness is essential for proving the intermediate-value and mean-value theorems (so does every other book, but it's done here in pedagogically simple style).
If you want
real books on analysis, you're spoilt for choice. Rudin's dated
Principles of Mathematical Analysis is still used at American universities (my boy used it a couple of years back). I'm reluctant to recommend it to quant students. One I would recommend is Bartle and Sherbert's
Introduction to Real Analysis, which is where I learnt my analysis from. Another I recommend is Bressoud's
A Radical Approach to Real Analysis: this is written by a master pedagogue who presents analysis through a historical framework. Why is this important? For beginners, it's not clear why we go through a careful construction of the real numbers, and the result that every Cauchy sequence converges in the reals: after all, the results that we eventually come to are already known from calculus (e.g. mean-value theorem, definition and basic properties of the Riemann integral): why bother proving all this obvious material? The reasons are rooted in history: how to prove that the ratio and root tests work, for example, or why every continuous function isn't differentiable. Our intuition leads us astray and hence the need for a sound foundation. Bressoud is ideal for explaining how and why the sound foundation came into being historically. So Bartle and Bressoud are a couple of recommendations. I may come up with one or two more.
The starting point is the completeness of the reals, and different ways of saying this (e.g., Dedekind cuts, convergence of Cauchy sequences, least upper bound). Then from completeness of the reals to various properties of sequences. Then the epsilon-delta definition of continuity.