Difference between revisions of "Maximum-minimum theorem"
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Assume if possible <math>\forall n\in\mathbb{N}\exists x_n\in [a,b]</math> such that <math>f(x_n)>n</math> | Assume if possible <math>\forall n\in\mathbb{N}\exists x_n\in [a,b]</math> such that <math>f(x_n)>n</math> | ||
− | As <math>[a,b]</math> is bounded, <math>\left\langle x_n\right\rangle</math> is bounded. | + | As <math>[a,b]</math> is bounded, <math>\left\langle x_n\right\rangle</math> is bounded. |
− | By the [[Bolzano-Weierstrass theorem]], there exists a sunsequence <math>\left\ | + | By the [[Bolzano-Weierstrass theorem]], there exists a sunsequence <math>\left\langle x_{n_r}\right\rangle</math> of <math>\left\langle x_n\right\rangle</math> which converges to <math>x</math>. |
As <math>[a,b]</math> is closed, <math>x\in [a,b]</math>. Hence, <math>f</math> is continous at <math>x</math>, and by the [[Limit|sequential criterion for limits]] <math>f(x_n)</math> is convergent, contradicting the assumption. | As <math>[a,b]</math> is closed, <math>x\in [a,b]</math>. Hence, <math>f</math> is continous at <math>x</math>, and by the [[Limit|sequential criterion for limits]] <math>f(x_n)</math> is convergent, contradicting the assumption. | ||
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i.e. <math>f(x)=M</math> | i.e. <math>f(x)=M</math> | ||
+ | |||
+ | ==References== | ||
+ | R.G. Bartle, D.R. Sherbert, <i>Introduction to Real Analysis</i> | ||
+ | |||
+ | ==See Also== | ||
+ | <UL> | ||
+ | <LI>Calculus</LI> | ||
+ | <LI>Bolzano's theorem</LI> | ||
+ | </UL> |
Revision as of 23:41, 14 February 2008
The Maximum-minimum theorem is a result about continous functions that deals with a property of intervals rather than that of the function itself.
Contents
Statement
Let
Let be continous on
Then, has an absolute maximum and an absolute minimum on
Proof
We will first show that is bounded on ...(1)
Assume if possible such that
As is bounded, is bounded.
By the Bolzano-Weierstrass theorem, there exists a sunsequence of which converges to .
As is closed, . Hence, is continous at , and by the sequential criterion for limits is convergent, contradicting the assumption.
Similarly we can show that is bounded below
Now, Let
By the Gap lemma, , such that
As is bounded, by Bolzano-Weierstrass theorem, has a subsequence that converges to
As is continous at ,
i.e.
References
R.G. Bartle, D.R. Sherbert, Introduction to Real Analysis
See Also
- Calculus
- Bolzano's theorem