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A sequence
(1) |
is called
A sequence which is either increasing, decreasing, non-increasing, or non-decreasing is called a monotone sequence. A sequence which is either increasing, or decreasing is called strictly monotone.
Theorem (The monotone convergence principle): (a) Let (1) be an increasing or non-decreasing sequence which is bounded above. Then (1) is a convergent sequence.
(b) Let (1) be an decreasing or non-increasing sequence which is bounded below. Then (1) is a convergent sequence.
(c) Let (1) be a monotone sequence which is bounded. Then (1) is a convergent sequence.
Example: Let be fixed and
(2) |
We have for every . We prove that if then the sequence , , is bounded above and thus convergent.
Consequently,
If we conclude that
(3) |
for every ,
Form estimation (3) there immediately follows that
(4) |
Cite this web-page as:
Štefan Porubský: Monotone convergence.