Calculus Suggestion for a proof-based math text book for high school

[CornMuffin]

I am currently teaching the pure mathematics II component of A-level mathematics which consist of the following subjects: Logarithmic/exponential functions, trigonometry, differentiation, integration (including integration by parts, and trig substitution), numerical approximation of roots of a function, vectors, differential equations, complex numbers.

My student asked me for a math book which goes through the proof of some of the properties, identities, equations and theorems or it's origin discussed in class. I often go through proofs in class, but the book that we use hardly ever have proofs.

Do you have any suggestion for a proof-based math book suitable for a curious, and above average high school math student.

Thank you

 

[micromass]

Lang - Basic mathematics is exactly what you're looking for.

 

[mathwonk]

For more depth and detail, have you looked at the calculus books by Spivak and Apostol?

 

[MidgetDwarf]

Maybe Axler, "Pre-calculus"? I found it better than Lang's Basic Mathematics.