Monotone convergence theorem

The monotone convergence theorem for real sequences states that every bounded monotone sequence converges. Its limit is the sequence’s supremum if it is increasing and its infimum if it is decreasing.

Monotone sequence: A sequence whose terms never decrease or never increase. The theorem applies precisely to sequences with this order pattern.
Increasing sequence: A sequence whose terms never decrease. Its terms approach their least upper bound when they have an upper bound.
Bolzano–Weierstrass theorem: Every bounded sequence in Euclidean space has a convergent subsequence. It guarantees subsequential convergence without requiring the entire sequence to be monotone.
Series convergence: The convergence of the sequence of partial sums of an infinite series. Monotonicity and boundedness of partial sums can prove a series converges.
Bernard Bolzano: A Czech mathematician and philosopher whose work anticipated major developments in analysis. His work helped establish rigorous treatments of limits and continuity that underpin sequence convergence.
Bounded sequence: A sequence whose terms all lie within fixed finite bounds. A monotone sequence needs a finite bound in the relevant direction to guarantee convergence.
Decreasing sequence: A sequence whose terms never increase. Its terms approach their greatest lower bound when they have a lower bound.
Cauchy convergence criterion: A sequence in the real numbers converges exactly when it is Cauchy. This criterion tests convergence through distances rather than order and bounds.
Nested interval theorem: A theorem stating that nested closed bounded intervals in the real numbers have a common point. Its proof can use monotone endpoint sequences and their limits.
Karl Weierstrass: A German mathematician who helped establish the rigorous foundations of mathematical analysis. His analytic rigor shaped the modern treatment of limits and convergence.

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