Nested Algebraic Symmetry in Inequality Proofs: A Variation on Cauchy's Classical Method

This work introduces a variation on Augustin-Louis Cauchy’s classical proof of the arithmetic–geometric mean (AM–GM) inequality as presented in his 'Cours d’analyse de l’École Royale Polytechnique'. While replacing Cauchy’s forward–backward induction with a recurrence-based argument, it retains and systematizes the nested symmetric algebraic structure that underlies the latter part of his original development. This symmetry-based approach extends naturally to other canonical inequalities, including Jensen’s, Minkowski’s, and — via mild algebraic elaboration — Hölder’s inequality.