Cauchy Sequences and Completeness

Sequences and Limit Theorems — Convergence, Subsequences & Completeness, done rigorously Every "clearly" and "evidently" in the theory of sequences, made visible. This is a complete, proof-driven tour of how sequences converge — built from the ε–N definition up to the Cauchy criterion and completeness, with every step animated and narrated. No skipped lines, no hand-waving. IN THIS VIDEO • The algebra of limits — sums, scalar multiples, products, and reciprocals, each proved from the definition and then checked on a worked example in exact fractions. • Sequences in higher dimensions — convergence coordinate by coordinate, and the arithmetic that follows, including dot products. • Subsequences — what they reveal, why a bounded sequence must hide a convergent one, and why the set of subsequential limits is always closed. • Cauchy sequences & completeness — the diameter viewpoint, nested sets, why Euclidean space is complete, and two spaces that are NOT: the rationals (with a hole at √2) and the open interval (0,1), where the limit escapes the space. • The Monotone Convergence Theorem — bounded plus monotone forces convergence, with the limit landing exactly on the supremum.