Difference between revisions of "Maximum-minimum theorem"

Revision as of 00:37, 15 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.

Statement

Let $f:[a,b]\rightarrow\mathbb{R}$

Let $f$ be continous on $[a,b]$

Then, $f$ has an absolute maximum and an absolute minimum on $[a,b]$

Proof

We will first show that $f$ is bounded on $[a,b]$ ...(1)

Assume if possible $\forall n\in\mathbb{N}\exists x_n\in [a,b]$ such that $f(x_n)>n$

As $[a,b]$ is bounded, $\left\langle x_n\right\rangle$ is bounded.

By the Bolzano-Weierstrass theorem, there exists a sunsequence $\left\langlex_{n_r}\right\rangle$ (Error compiling LaTeX. Unknown error_msg) of $\left\langle x_n\right\rangle$ which converges to $x$ .

As $[a,b]$ is closed, $x\in [a,b]$ . Hence, $f$ is continous at $x$ , and by the sequential criterion for limits $f(x_n)$ is convergent, contradicting the assumption.

Similarly we can show that $f$ is bounded below

Now, Let $M=\sup\{f([a,b])\}$

By the Gap lemma, $\forall n\in\mathbb{N}$ , $\exists x_n$ such that $M-f(x_n)<\frac{1}{n}$

As $\left\langle x_n\right\rangle$ is bounded, by Bolzano-Weierstrass theorem, $\left\langle x_n\right\rangle$ has a subsequence $\left\langle x_{n_r}\right\rangle$ that converges to $x\in [a,b]$

As $f$ is continous at $x$ , $f(x)\in V_{\frac{1}{n}}(M)\forall n$

i.e. $f(x)=M$

@@ Line 9: / Line 9: @@
 ==Proof==
+We will first show that <math>f</math> is bounded on <math>[a,b]</math>...(1)
+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.
+By the [[Bolzano-Weierstrass theorem]], there exists a sunsequence <math>\left\langlex_{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.
+Similarly we can show that <math>f</math> is bounded below
+Now, Let <math>M=\sup\{f([a,b])\}</math>
+By the [[Gap lemma]], <math>\forall n\in\mathbb{N}</math>, <math>\exists x_n</math> such that <math>M-f(x_n)<\frac{1}{n}</math>
+As <math>\left\langle x_n\right\rangle</math> is bounded, by [[Bolzano-Weierstrass theorem]], <math>\left\langle x_n\right\rangle</math> has a subsequence <math>\left\langle x_{n_r}\right\rangle</math> that converges to <math>x\in [a,b]</math>
+As <math>f</math> is continous at <math>x</math>, <math>f(x)\in V_{\frac{1}{n}}(M)\forall n</math>
+i.e. <math>f(x)=M</math>

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Difference between revisions of "Maximum-minimum theorem"

Revision as of 00:37, 15 February 2008

Statement

Proof