This paper investigates the multivariate generalization of Cauchy's inequality 1 + x ≤ ex, where x is any non-negative real number. Specifically, it aims to prove the inequality (1 + x1)(1 + x2)...(1 + xn) ≤ e(x1+x2+...+xn), where x1, x2, ..., xn are pairwise distinct non-negative real numbers. The proof is based on notions from empty product conventions and Beppo Levi's theorem of monotone convergence. This inequality is also extended to simultaneous inequalities and its relationship to ordinary differential equation Cauchy problems and population dynamics is explored. Direct approaches using definitions of monotone functions and mean value theore