Monotone Convergence Theorem
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Contents |
[edit] Theorem
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 B.
We need to show that
as
.
Let ε > 0.
Since B − ε is not an upper bound, by the definition of supremum.
Thus
.
But
is increasing.
Hence
.
But B is still an upper bound for
.
| < |
| ||||
|
| < | xn < B + ε | Real Plus Epsilon | ||
|
| < | ε | 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.

