The key property of the reals, the fundamental axiom which makes everything work, can be stated as follows:
The fundamental axiom of analysis. If an £ R for each n > 1, A 6 R and a\ < a2 < аз < ... and an < A for each n, then there exists an a 6K such that an —> a as n —> oo.
Less ponderously, and just as rigorously, the fundamental axiom for the real numbers says every increasing sequence bounded above tends to a limit.
Everything which depends on the fundamental axiom is analysis, everything else is mere algebra.
I claim that all the theorems of classical analysis can be obtained from the standard 'algebraic' properties of R together with the fundamental axiom. I shall start by trying to prove the intermediate value theorem. (Here [a, b] is the closed interval [a, b] = {x 6 R : a < x <b}.)