Is there a problem in studying analysis before calculus?

I have always held that the best way to learn mathematics is by doing computations and working out concrete examples, thus taking an abstract theorem based course without the intuition gained from working out the motivating examples I think in general is not a good idea. Proving theorems will improve your mathematical reasoning and maturity, but generally speaking it doesn't help you solve problems. Once you've gained a feel for things based on the examples you have worked out you gain more of an intuition for proving deeper theorems as well. Moreover, I took analysis years ago and I don't remember anything, yet basic calculus comes up all the time (even in my research).

In most American universities, Real Analysis subsumes Calculus. Therefore, there is no loss in skipping straight to Real Analysis-- provided that you can handle it, i.e. provided that you have the "mathematical maturity."

I personally skipped calculus, and the only problem I ran into was the following: I found that people who took calculus tend to develop proficiency at problem solving techniques that I didn't develop during analysis. Thus, they had lots of homework sets that covered routine problems which required them to learn problem solving techniques, whereas analysis was focused more on "theorem proving" and not as much on "problem solving." However, this is easy enough to solve. Just crack open a calculus textbook and do a hundred or so exercises.

They are basically the same subject, but if you cannot judge whether a series converges, how to do a double integral, how to apply Stoke's formula, etc, then you may benefit some studying some basics first. After all analysis is in place to make calculus rigorous, and if you do not know calculus well, chances are if you study higher level analysis you will get into numerous difficulties because they are all based on calculus. For example in elementary ODE, PDE, complex analysis, etc. I do not see the point to rush studying a subject.

A good mathematical analysis text to recommend is Zorich's book. The book I is more elementary and may be substituted by Rudin if you wish.