While Courant's book isn't as rigorous as Spivak's, it's certainly more rigorous than any of the more popular undergraduate texts. Stewart, for instance, is merely bearable: most of the proofs are there, dutifully interlarded, but they're generally useless with regard to the end-of-chapter...
Differential and Integral Calculus Vol I. -- Richard Courant
Differential and Integral Calculus Vol II. -- Richard Courant
Calculus with Analytic Geometry, 2nd Ed. -- George F. Simmons
Calculus and Analytic Geometry, 4th Ed. -- George B. Thomas
Linear Algebra -- Jim Hefferon