|
|
|
|
![[parent]](http://images.planetmath.org:8080/images/uparrow.png)
limit of nondecreasing sequence
|
(Theorem)
|
|
|
Theorem. A monotonically nondecreasing sequence of real numbers with upper bound a number $M$ converges to a limit which does not exceed $M$ .
Proof. Let $a_1 \leqq a_2 \leqq \ldots \leqq a_n \leqq \ldots \leqq M$ . Therefore the set $\{a_1,\,a_2,\,\ldots\}$ has a finite supremum $s \leqq M$ . We show that
 |
(1) |
Let $\varepsilon$ an arbitrary positive number. According to the definition of supremum we have $a_n \leqq s$ for all $n$ and on the other hand, there exists a member $a_{n(\varepsilon)}$ of the sequence that is $> s-\varepsilon$ . Then we have $s-\varepsilon < a_{n(\varepsilon)} \leqq s$ , and since the sequence is nondecreasing, $$0 \leqq s-a_n \leqq s\!-\!a_{n(\varepsilon)} < \varepsilon\quad \mbox{for all}\;\, n \geqq n(\varepsilon).$$ Thus the equation (1) and the whole theorem has been proven.
For the nonincreasing sequences there is the corresponding
Theorem. A monotonically nonincreasing sequence of real numbers with lower bound a number $L$ converges to a limit which is not less than $L$ .
Note. A good application of the latter theorem is in the proof that Euler's constant exists.
|
"limit of nondecreasing sequence" is owned by pahio.
|
|
(view preamble | get metadata)
Cross-references: proof that Euler's constant exists, application, lower bound, monotonically nonincreasing, equation, positive, supremum, finite, proof, limit, converges, number, upper bound, real numbers, sequence, monotonically nondecreasing, theorem
There are 4 references to this entry.
This is version 9 of limit of nondecreasing sequence, born on 2007-12-08, modified 2008-01-04.
Object id is 10114, canonical name is NondecreasingSequenceWithUpperBound.
Accessed 2214 times total.
Classification:
| AMS MSC: | 40-00 (Sequences, series, summability :: General reference works ) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|
