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In mathematics, specifically in [[real analysis]], the '''Bolzano–Weierstrass theorem''', named after [[Bernard Bolzano]] and [[Karl Weierstrass]], is a fundamental result about convergence in a finite-dimensional [[Euclidean space]] '''R'''<sup>''n''</sup>. The theorem states that
each [[bounded sequence]] in '''R'''<sup>''n''</sup> has a [[limit of a sequence|convergent]] [[subsequence]]. An equivalent formulation is that a subset of '''R'''<sup>''n''</sup> is [[Sequentially compact space|sequentially compact]] if and only if it is [[closed set|closed]] and [[bounded set|bounded]].
 
== Proof ==
 
First we prove the theorem when ''n''&nbsp;= 1, in which case the ordering on '''R''' can be put to good use. Indeed we have the following result.
 
'''Lemma''': Every sequence {{nowrap begin}}{&thinsp;''x''<sub>''n''</sub>&thinsp;}{{nowrap end}} in '''R''' has a [[monotone sequence|monotone]] [[subsequence]].
 
'''Proof''': Let us call a positive integer ''n'' a "'''peak''' of the sequence" if ''m''&nbsp;> ''n'' implies&thinsp; {{nowrap|''x''<sub>&thinsp;''n''</sub> > ''x''<sub>&thinsp;''m''</sub>}}&thinsp; ''i.e.'', if&thinsp; ''x''<sub>''n''</sub> is greater than every subsequent term in the sequence. Suppose first that the sequence has infinitely many peaks, ''n''<sub>1</sub>&nbsp;< ''n''<sub>2</sub>&nbsp;< ''n''<sub>3</sub>&nbsp;<&nbsp;…&nbsp;< ''n''<sub>''j''</sub>&nbsp;<&nbsp;…. Then the subsequence&thinsp; <math> \{x_{n_j}\}</math>&thinsp; corresponding to these peaks is monotonically decreasing, and we are done. So suppose now that there are only finitely many peaks, let ''N'' be the last peak and {{nowrap|''n''<sub>1</sub> {{=}} ''N'' + 1}}. Then ''n''<sub>1</sub> is not a peak, since {{nowrap|''n''<sub>1</sub> > ''N''}}, which implies the existence of an {{nowrap|''n''<sub>2</sub> > ''n''<sub>1</sub>}} with&nbsp; <math>x_{n_2} \geq x_{n_1}.</math>&thinsp;  Again, {{nowrap|''n''<sub>2</sub> > ''N''}} is not a peak, hence there is {{nowrap|''n''<sub>3</sub> > ''n''<sub>2</sub>}} with <math>x_{n_3} \geq x_{n_2}.</math>&thinsp;  Repeating this process leads to an infinite non-decreasing  subsequence&nbsp; <math>x_{n_1} \leq x_{n_2} \leq x_{n_3} \leq \ldots</math>, as desired.
 
Now suppose we have a [[bounded sequence]] in '''R'''; by the Lemma [[there exists]] a monotone subsequence, necessarily bounded. It follows from the [[monotone convergence theorem]] that this subsequence must converge.
 
Finally, the general case can be easily reduced to the case of ''n''&nbsp;= 1 as follows: given a bounded sequence in '''R'''<sup>''n''</sup>, the sequence of first coordinates is a bounded real sequence, hence has a convergent subsequence. We can then extract a subsubsequence on which the second coordinates converge, and so on, until in the end we have passed from the original sequence to a subsequence ''n'' times &mdash; which is still a subsequence of the original sequence &mdash; on which each coordinate sequence converges, hence the subsequence itself is convergent.
 
== Sequential compactness in Euclidean spaces ==
 
Suppose ''A'' is a subset of '''R'''<sup>''n''</sup> with the property that every sequence in ''A'' has a subsequence converging to an element of ''A''.  Then ''A'' must be bounded, since otherwise there exists a sequence ''x''<sub>''m''</sub> in ''A'' with  {{nowrap begin}}||&thinsp;''x''<sub>''m''</sub>&thinsp;|| ≥ ''m''{{nowrap end}} for all ''m'', and then every subsequence is unbounded and therefore not convergent. Moreover ''A'' must be closed, since from a noninterior point ''x'' in the complement of ''A'' one can build an ''A''-valued sequence converging to ''x''. Thus the subsets ''A'' of '''R'''<sup>''n''</sup> for which every sequence in ''A'' has a subsequence converging to an element of ''A'' &ndash;&nbsp;i.e., the subsets which are [[sequentially compact]] in the subspace topology&nbsp;&ndash; are precisely the closed and bounded sets.
 
This form of the theorem makes especially clear the analogy to the [[Heine–Borel theorem]],  
which asserts that a subset of '''R'''<sup>''n''</sup> is compact if and only if it is closed and bounded. In fact, general topology tells us that a metrizable space is compact if and only if it is sequentially compact, so that the Bolzano–Weierstrass and Heine–Borel theorems are essentially the same.
 
== History ==
 
The Bolzano–Weierstrass theorem is named after mathematicians [[Bernard Bolzano]] and [[Karl Weierstrass]].  It was actually first proved by Bolzano in 1817 as a [[Lemma (mathematics)|lemma]] in the proof of the [[intermediate value theorem]].  Some fifty years later the result was identified as significant in its own right, and proved again by Weierstrass. It has since become an essential theorem of [[Real analysis|analysis]].
 
== Application to economics ==
There are different important [[economic equilibrium|equilibrium]] concepts in economics, the proofs of the existence of which often require variations of the Bolzano–Weierstrass theorem. One example is the existence of a [[Pareto efficiency|Pareto efficient]] allocation. An allocation is a matrix of consumption bundles for agents in an economy, and an allocation is Pareto efficient if no change can be made to it which makes no agent worse off and at least one agent better off (here rows of the allocation matrix must be rankable by a [[preference relation]]). The Bolzano–Weierstrass theorem allows one to prove that if the set of allocations is compact and non-empty, then the system has a Pareto-efficient allocation.
 
==See also==
*[[Sequentially compact space]]
*[[Heine–Borel theorem]]
*[[Fundamental axiom of analysis]]
 
== References ==
#{{note|Fitzpatrick}} Fitzpatrick, Patrick M. (2006) Advanced Calculus (2nd ed.). Belmont, CA: Thompson Brooks/Cole. ISBN 0-534-37603-7.
 
==External links==
* {{springer|title=Bolzano-Weierstrass theorem|id=p/b016880}}
* [http://ram.rachum.com/bw.htm A proof of the Bolzano–Weierstrass theorem]
* [http://planetmath.org/?op=getobj&from=objects&id=2129 PlanetMath: proof of Bolzano–Weierstrass Theorem]
* [http://www.youtube.com/watch?v=dfO18klwKHg A proof of the Bolzano–Weierstrass theorem as a rap]
* [http://www.nakedprogrammer.com/BalzanoWeierstrass.html  A demonstration of the Bolzano–Weierstrass theorem]
 
{{DEFAULTSORT:Bolzano-Weierstrass theorem}}
[[Category:Theorems in real analysis]]
[[Category:Compactness theorems]]

Latest revision as of 16:27, 7 January 2015

In case you adore Nike shoes,. SB: abbreviation connected with skateboard, the negotiator shoes is Nike DUNK SB.
AP:,. LE: limited format.. ND: new model, introducing new elements inside the original design (mostly represents use the different materials, such because patent leather, fine mesh fabric, anti-skin......)TB: organization bulk,. cheap nike shoes.NU: inspired through the original air The nike jordan shoes when The nike jordan brand was re-carving footwear, like Nu popular.

RETRO: re-carved fashion, air Jordan was just about the most re-carved shoes. FLA: special color release of footlockerFNL: distinctive color edition connected with finish line
T: back,,, sometimes, it is definitely marked with P. Sample: as the news goes, it is really a sample of certain set of Nike shoes and reference too. And only intended for internal use, to some degree, it is THE. SC: sports basic,.

Prototype: trial model of shoes:. nike oem shoes china.Picture sample:. TB: company basketball: designed particularly for the team. Air flow Force 1: it is a so-called AF1, just as typically the name implies this is really a kind associated with shoes containing environment principles.
AIR MAXIMUM: to adapt in the tastes of transforming market, Nike unveiled AIR MAX line with transparent additional air cushion which unfortunately booms the getting of Nike OXYGEN MAX shoes. ACG: most conditions gear,Nike once again introduced a type of shoes applied along with a warm and watertight outer fabric, of which this is ACG.

100 % FREE: a nike nz newly revealed type, take full greatest things about the advanced technology to create consumer a much more flexible and comfortable feeling plus experience.nike original cheap.
. They are the goods of partnerships through professional athletes, activities medicine experts, and even designers. cheap nike oem shoes.This is all to your purpose to create a shoe that but not only looks good but can do effectively as another important camera playing basketball.

Basketball shoes also provide a dynastic level of quality. They build relating to the successes and downfalls of previous versions and add new innovations from the own. A great example may be the cheap nike Air Jordan a list of shoes.. This is exactly why the Nike Focus BB II has reviewed.
The Zoom BB can be a staple shoe for just about any serious player being exercised by teams in both NBA and NCAA school teams. So you know it is a show that is tried and tested in numerous playing conditions. The shoe's develop is pretty self-explanatory., high ankle, support and regarding it's cushioning this Zoom Air unit from the heel and foot...

It has holes cut from it to maximize this flow of air from the shoe allowing your toes to breath and also letting moisture right from sweat out. The shoe furthermore uses the ever before functional high top design to give good ankle support and forestall rollovers which are among the most common sources regarding basketball injuries.

Nonetheless it forgoes the firm wrap around structure useful to provide additional steadiness.cheap nike original shoes.Time will see if moving in this direction was a good plan for Nike... The shoe does have several negatives. The heel coup as an example is designed a trifle too large.

nike nz To counter this particular, a player includes two wear to help pairs of socks. High quality sports shoe carries a comfortable fit.. Other than that that is a pretty consistent sneaker.
Internet is probably the greatest tools that has become introduced on the planet markets for the several purposes that vary from seeking information, to help sharing thoughts to help shopping etc.,. All those who are eager for save the holiday to the markets and also exhausting themselves can makes use of the Nike size index charts,.

One of the top places to get the Nike shoes online is a websites of the particular authorized Nike shoes and boots dealers. One can makes use of the powerful search applications to generate a directory of the genuine Nike seller websites offering shipping and delivery on their city or your vicinities.

One could also sift through the variety of the reputed web based shoes retails which regularly buy the Nike stock options in bulk and gives better bargains on their online customers.. In order to locate the genuine and reliable Nike shoes one has to ensure to buy through the reputed online Nike retails as well as dealers.

If question any doubts around the product they ought not buy the Nike shoes through the virtual markets, rather invest amount of time in researching about the shoes they want and buy it from your real market.
With the aid of the introduction from the Nike ID practice, it is at this time even possible to have personalized shoes belonging to the Nike retails. All you've to do, is without a doubt specify their measurements, the material as well as color preferences, opted for trendy style and have absolutely their shoes delivered in the home within the given time.

However, when buying online one should make sure to learn to read the stipulations related to a aspects like shipping, payment, guarantees, warranties together with other related issues. All those who may be hesitant in purchasing the Nike shoes from virtual reality, due to possibility of scamming, who keep an amount of Nike...
buy cheap nike shoes online.However, when it concerns buying the trainers, always make sure to get them during the evening so the feet are inside their normal size from a day's exertion.

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