Dedekind Continuum Theorem

The simple but seminal notions of tangency and quadrature of a curve in Geometry evolved over the centuries into Infinitesimal Calculus, an extremely useful mathematical domain but with unsound logical foundations. Dedekind Continuum Theorem's significance lies in allowing a logically rigorous re-engineering of Calculus into Real Analysis based on the limit concept using the order-completeness of the real number system. The Theorem’s consequences – Bolzano-Weierstrass Theorem, Least Upper Bound Theorem, Heine-Borel Theorem, etc.,- enable tackling the infinitary processes of convergence of sequences and infinite series, as well as establishing central propositions about continuity, differentiability, and integrability of functions, ranging from Intermediate Value Theorem to Differentiation Rules Proposition to Rolle’s Theorem to Taylor’s Theorem to the Fundamental Theorem of Calculus. An elementary proof is offered for the closed expression for Riemann zeta function, ζ(2k) = ∑_(n=1)^∞▒〖1/n〗2k, k a positive integer.