Need a theoretical textbook for calculus, proof based

Question

The course descriptions is : A theoretical course in calculus; emphasizing proofs and techniques. Trigonometric identities. Limits and continuity; least upper bounds, intermediate and extreme value theorems. Derivatives, mean value and inverse function theorems. Integrals; fundamental theorem; elementary transcendental functions. Techniques of integration. Taylor's theorem; sequences and series; uniform convergence and power series.

This course is using Spivak's calculus, but I'm also looking for another proof based book accompany with Spivak, thx

Answer 1

The Theory of The Calculus is called Mathematical Analysis ...Elementary Analysis for the R to R functions....so any of the very many books on Analysis should serve your purpose.

Answer 2

You might try the standards:

1. Apostol - Calculus Vol I
2. Hardy - A Course of Pure Mathematics
3. Piskunov - Differential and Integral Calculus

You might also try Rosenlicht's "Introduction to Analysis". It's a cheap, awesome Dover book that presents basic analysis at a good level. Not exactly parallel to Spivak, but you might find the presentation enlightening paired with Spivak.