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| | The level of popularity of friv video games is mainly thanks to its utilization of intelligent concepts, most current graphics, engineering, on line flash update and educative benefit.<br><br>The games look attractive to mother and father as effectively as kids because of the moment detailing and very good story line. Mom and dad simply just love the strategy of training the tact to fix the difficulties and overcome challenges to their wards via these games. You will be thrilled and enthused to reach the upcoming degree of the activity and facial area the problem in a sporty way.<br><br>Soon after all everyday living lived with an attitude of sportsman helps everybody to come to be a dilemma solver.<br>The Pc or private laptop is a vital requirement in each and every portion of our lives. Business enterprise, conversation, instruction, athletics, entertainment and browsing are many spots that laptop has protected. Turning the wise Laptop into perform station or gaming zone is wise move for the techies.<br>Unlimited internet accessibility has aided to transform online gaming simpler and attainable. Individuals are cozy to benefit from a flash or Java site and enjoy their favored recreation at friv gaming zones. Major engineers and experts in the discipline are busy creating far more highly developed and consumer helpful variations each and every day.<br><br>Latest titles are all stacked up on the internet sites for the players to click their selection. Now gamers are spoilt for option with so many option online games available to them. You will come across thousand of renowned titles underneath many genres offered on the net now. Disentangling and enjoyable the minds are two important benefits of on the web online games.<br>Moreover personal perceptions like good tuning the reflexes and producing interactive mind-set are the track record benefits of participating in on the internet video games. Friv game titles assist to sharpen the blunt intellects of all people with masses of entertainment price.<br><br>FrivMe.com updates and provides latest flash online games of wide categorizes. Check out out the most recent selection of friv video games at FrivMe.com to locate amazing titles waiting around for yo u to discover. Have interaction in the successful pastime and get away from the monotonous mundane regime.<br><br>If you loved this information and you want to receive details concerning [http://freeonlinegames322099.pen.io/ Friv Online Games] kindly visit our web-page. |
| 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
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| 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]].
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| == Proof ==
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| First we prove the theorem when ''n'' = 1, in which case the ordering on '''R''' can be put to good use. Indeed we have the following result.
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| '''Lemma''': Every sequence {{nowrap begin}}{ ''x''<sub>''n''</sub> }{{nowrap end}} in '''R''' has a [[monotone sequence|monotone]] [[subsequence]].
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| '''Proof''': Let us call a positive integer ''n'' a "'''peak''' of the sequence" if ''m'' > ''n'' implies  {{nowrap|''x''<sub> ''n''</sub> > ''x''<sub> ''m''</sub>}}  ''i.e.'', if  ''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> < ''n''<sub>2</sub> < ''n''<sub>3</sub> < … < ''n''<sub>''j''</sub> < …. Then the subsequence  <math> \{x_{n_j}\}</math>  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 <math>x_{n_2} \geq x_{n_1}.</math>  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>  Repeating this process leads to an infinite non-decreasing subsequence <math>x_{n_1} \leq x_{n_2} \leq x_{n_3} \leq \ldots</math>, as desired.
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| 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.
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| Finally, the general case can be easily reduced to the case of ''n'' = 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 — which is still a subsequence of the original sequence — on which each coordinate sequence converges, hence the subsequence itself is convergent.
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| == Sequential compactness in Euclidean spaces ==
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| 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}}|| ''x''<sub>''m''</sub> || ≥ ''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'' – i.e., the subsets which are [[sequentially compact]] in the subspace topology – are precisely the closed and bounded sets.
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| This form of the theorem makes especially clear the analogy to the [[Heine–Borel theorem]],
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| 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.
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| == History ==
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| 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]].
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| == Application to economics ==
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| 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.
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| ==See also==
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| *[[Sequentially compact space]]
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| *[[Heine–Borel theorem]]
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| *[[Fundamental axiom of analysis]]
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| == References ==
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| #{{note|Fitzpatrick}} Fitzpatrick, Patrick M. (2006) Advanced Calculus (2nd ed.). Belmont, CA: Thompson Brooks/Cole. ISBN 0-534-37603-7.
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| ==External links==
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| * {{springer|title=Bolzano-Weierstrass theorem|id=p/b016880}}
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| * [http://ram.rachum.com/bw.htm A proof of the Bolzano–Weierstrass theorem]
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| * [http://planetmath.org/?op=getobj&from=objects&id=2129 PlanetMath: proof of Bolzano–Weierstrass Theorem]
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| * [http://www.youtube.com/watch?v=dfO18klwKHg A proof of the Bolzano–Weierstrass theorem as a rap]
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| * [http://www.nakedprogrammer.com/BalzanoWeierstrass.html A demonstration of the Bolzano–Weierstrass theorem]
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| {{DEFAULTSORT:Bolzano-Weierstrass theorem}}
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| [[Category:Theorems in real analysis]]
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| [[Category:Compactness theorems]]
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The level of popularity of friv video games is mainly thanks to its utilization of intelligent concepts, most current graphics, engineering, on line flash update and educative benefit.
The games look attractive to mother and father as effectively as kids because of the moment detailing and very good story line. Mom and dad simply just love the strategy of training the tact to fix the difficulties and overcome challenges to their wards via these games. You will be thrilled and enthused to reach the upcoming degree of the activity and facial area the problem in a sporty way.
Soon after all everyday living lived with an attitude of sportsman helps everybody to come to be a dilemma solver.
The Pc or private laptop is a vital requirement in each and every portion of our lives. Business enterprise, conversation, instruction, athletics, entertainment and browsing are many spots that laptop has protected. Turning the wise Laptop into perform station or gaming zone is wise move for the techies.
Unlimited internet accessibility has aided to transform online gaming simpler and attainable. Individuals are cozy to benefit from a flash or Java site and enjoy their favored recreation at friv gaming zones. Major engineers and experts in the discipline are busy creating far more highly developed and consumer helpful variations each and every day.
Latest titles are all stacked up on the internet sites for the players to click their selection. Now gamers are spoilt for option with so many option online games available to them. You will come across thousand of renowned titles underneath many genres offered on the net now. Disentangling and enjoyable the minds are two important benefits of on the web online games.
Moreover personal perceptions like good tuning the reflexes and producing interactive mind-set are the track record benefits of participating in on the internet video games. Friv game titles assist to sharpen the blunt intellects of all people with masses of entertainment price.
FrivMe.com updates and provides latest flash online games of wide categorizes. Check out out the most recent selection of friv video games at FrivMe.com to locate amazing titles waiting around for yo u to discover. Have interaction in the successful pastime and get away from the monotonous mundane regime.
If you loved this information and you want to receive details concerning Friv Online Games kindly visit our web-page.