Is there a calculus (not analysis, not second-course in calculus) textbook that begins with sequences and series before moving on to functions of a real variable? I'm looking for a good dose of rigor too -- for instance, epsilon-delta definitions make an appearance but don't dominate the exposition, as is the style in most "honors calculus" books -- but probably any book that starts with sequences will have this.
It seems to me the main reasons calculus books start with limits of functions and then derivatives are inertia and curriculum constraints: Calc I is the terminal math course taken by some majors -- maybe biology, maybe business, etc. -- and that's they content those majors want.
I'm teaching a rigorous calculus course for math majors (but designed for students coming straight out of US high school, many with no prior calculus experience -- the average problem difficulty in Spivak seems a little too high, although the exposition is beautiful). I have the students for a whole year, so I'd really like to start them with sequences and series and move to functions of a real variable later. This is the more natural sequencing. But more importantly, students who have some prior calculus exposure will hopefully stay more engaged, or if not, at least realize immediately that they will need to get to work and cannot coast along on previous knowledge.