boundedness theorem

Boundedness Theorem. Let a and b be real numbers with a<b, and let f be a continuousMathworldPlanetmathPlanetmath, real valued function on [a,b]. Then f is bounded above and below on [a,b].

Proof. Suppose not. Then for all natural numbersMathworldPlanetmath n we can find some xn[a,b] such that |f(xn)|>n. The sequenceMathworldPlanetmath (xn) is boundedPlanetmathPlanetmathPlanetmathPlanetmath, so by the Bolzano-Weierstrass theoremMathworldPlanetmath it has a convergentMathworldPlanetmathPlanetmath sub sequence, say (xni). As [a,b] is closed (xni) convergesPlanetmathPlanetmath to a value in [a,b]. By the continuity of f we should have that f(xni) converges, but by construction it diverges. This contradictionMathworldPlanetmathPlanetmath finishes the proof.

Title boundedness theorem
Canonical name BoundednessTheorem
Date of creation 2013-03-22 14:29:18
Last modified on 2013-03-22 14:29:18
Owner classicleft (5752)
Last modified by classicleft (5752)
Numerical id 6
Author classicleft (5752)
Entry type Theorem
Classification msc 26A06