Monotone convergence theorem

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In mathematics, there are several theorems dubbed monotone convergence; here we present some major examples.

Contents

Convergence of a monotone sequence of real numbers

Theorem

If ak is a monotone sequence of real numbers (e.g., if ak ≤ ak+1,) then this sequence has a finite limit if and only if the sequence is bounded.[1]

Proof

We prove that if an increasing sequence {an} is bounded above, then it is convergent and the limit is \sup_n \{a_n\}.

Since {an} is non-empty and by assumption, it is bounded above, therefore, by the Least upper bound property of real numbers, c = \sup_n \{a_n\} exists and is finite. Now for every \varepsilon > 0, there exists aN such that a_N > c - \varepsilon , since otherwise c - \varepsilon is an upper bound of {an}, which contradicts to c being \sup_n \{a_n\}. Then since {an} is increasing, \forall n > N , |c - a_n| = c - a_n \leq c - a_N < \varepsilon , hence by definition, the limit of {an} is \sup_n \{a_n\}.

Remark

If a sequence of real numbers is decreasing and bounded below, then its infimum is the limit.

Convergence of a monotone series

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