PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (204)
Orphanage
Unclass'd
Unproven (490)
Corrections (7)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: Very high Entry average rating: No information on entry rating
[parent] 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

$\displaystyle \lim_{n\to\infty}a_n = s.$ (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,
$\displaystyle 0 \leqq s-a_n \leqq s\!-\!a_{n(\varepsilon)} < \varepsilon$   for all$\displaystyle \;\, n \geqq n(\varepsilon).$
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)

View style:

See Also: monotonically increasing, monotone increasing, supremum, infimum, convergence of the sequence (1+1/n)^n

Other names:  nondecreasing sequence with upper bound, limit of increasing sequence
Keywords:  nondecreasing, bounded from above, nonincreasing, bounded from below

This object's parent.

Attachments:
example of converging increasing sequence (Example) by pahio
Log in to rate this entry.
(view current ratings)

Cross-references: proof that Euler's constant exists, lower bound, monotonically nonincreasing, equation, positive, supremum, finite, limit, converges, number, upper bound, real numbers, sequence, monotonically nondecreasing
There are 2 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 609 times total.

Classification:
AMS MSC: 40-00 (Sequences, series, summability :: General reference works )
Pending Errata and Addenda
None.
Discussion
Style: Expand: Order:
forum policy
Nondecreasing sequence with upper bound by pahio on 2007-12-08 15:47:45
I did not find this important theorem in PM, and therefore I wrote this entry. It were good if this entry had alternative titles -- for that one could more easily find it. Can someone propose alternative titles?
Jussi
[ reply | up ]
Interact
post | correct | update request | prove | add result | add corollary | add example | add (any)