The least-upper-bound principle The subject of real analysis can be founded upon the least-upper-bound principle, a version of Dedekind completeness. Taking this as a core principle, one proceeds to prove all the familiar foundational theorems, such as the intermediate value theorem, the Heine-Borel theorem and many others. In a sense, the least-upper-bound principle is to real analysis what the induction principle is to number theory.
I first learned of continuous induction from Pete Clark, who has a very nice account of it at: The instructor's guide to real induction (http://alpha.math.uga.edu/~pete/instructors_guide_2017.pdf).