Another text that many people gush over is G. H. Hardy's A Course of Pure Mathematics. I've never read it, but at least a couple of editions of it are in the public domain. (Check archive.org.)
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
Linear Algebra...