Monotone Convergence Theorem

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Contents 1 Theorem 1.1 Increasing Sequence 1.2 Decreasing Sequence 2 Proof 2.1 Proof for Increasing Sequence 2.2 Proof for Decreasing Sequence 3 Sources

[edit] Theorem

Every bounded monotone sequence is convergent.

Let be a sequence in .

[edit] Increasing Sequence

Let be increasing and bounded above.

Then converges to its supremum.

[edit] Decreasing Sequence

Let be decreasing and bounded below.

Then converges to its infimum.

[edit] Proof

[edit] Proof for Increasing Sequence

Suppose is increasing and bounded above.

Let its supremum be .

We need to show that as .

Let .

Since is not an upper bound, by the definition of supremum.

Thus .

But is increasing.

Hence .

But is still an upper bound for .

					Real Plus Epsilon
					Negative of Absolute Value: Corollary

Hence the result.

[edit] Proof for Decreasing Sequence

If is decreasing and bounded below then is increasing and bounded above.

Thus the above result applies and the proof follows.

[edit] Sources

• W.A. Sutherland: Introduction to Metric and Topological Spaces (1975): Theorem
• K.G. Binmore: Mathematical Analysis: A Straightforward Approach (1977):