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 a_{k} is a monotone sequence of real numbers (e.g., if a_{k} ≤ a_{k+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 {a_{n}} is bounded above, then it is convergent and the limit is .
Since {a_{n}} is nonempty and by assumption, it is bounded above, therefore, by the Least upper bound property of real numbers, exists and is finite. Now for every , there exists a_{N} such that , since otherwise is an upper bound of {a_{n}}, which contradicts to c being . Then since {a_{n}} is increasing, , hence by definition, the limit of {a_{n}} is
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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