You're leaving out a fairly large middle ground of boring medium-length proofs. See, e.g. construction of the reals from the rationals using Dedekind cuts in Ru

You're leaving out a fairly large middle ground of boring medium-length proofs. See, e.g. construction of the reals from the rationals using Dedekind cuts in Rudin's book: http://en.wikibooks.org/wiki/Real_Analysis/Dedekinds_construction http://en.wikibooks.org/wiki/Real_Analysis/Dedekinds_constru....