Monotone convergence theorem

Monotone convergence theorem In the mathematical field of real analysis, the monotone convergence theorem is any of a number of related theorems proving the convergence of monotonic sequences (sequences that are non-decreasing or non-increasing) that are also bounded. Informally, the theorems state that if a sequence is increasing and bounded above by a supremum, then the sequence will converge to the supremum; in the same way, if a sequence is decreasing and is bounded below by an infimum, it will converge to the infimum.

Contents 1 Convergence of a monotone sequence of real numbers 1.1 Lemma 1 1.2 Proof 1.3 Lemma 2 1.4 Proof 1.5 Theorem 1.6 Proof 2 Convergence of a monotone series 2.1 Theorem 3 Beppo Levi's lemma 3.1 Theorem 3.2 Proof 3.2.1 Intermediate results 3.2.1.1 Lebesgue integral as measure 3.2.1.1.1 Proof 3.2.1.2 "Continuity from below" 3.2.2 Proof of theorem 4 See also 5 Notes Convergence of a monotone sequence of real numbers Lemma 1 If a sequence of real numbers is increasing and bounded above, then its supremum is the limit.

Proof Let {displaystyle (a_{n})_{nin mathbb {N} }} be such a sequence, and let {displaystyle {a_{n}}} be the set of terms of {displaystyle (a_{n})_{nin mathbb {N} }} . By assumption, {displaystyle {a_{n}}} is non-empty and bounded above. By the least-upper-bound property of real numbers, {textstyle c=sup _{n}{a_{n}}} exists and is finite. Now, for every {displaystyle varepsilon >0} , there exists {displaystyle N} such that {displaystyle a_{N}>c-varepsilon } , since otherwise {displaystyle c-varepsilon } is an upper bound of {displaystyle {a_{n}}} , which contradicts the definition of {displaystyle c} . Then since {displaystyle (a_{n})_{nin mathbb {N} }} is increasing, and {displaystyle c} is its upper bound, for every {displaystyle n>N} , we have {displaystyle |c-a_{n}|leq |c-a_{N}|

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