proof of intermediate value theorem

We first prove the following lemma.

If f:[a,b] is a continuous functionMathworldPlanetmathPlanetmath with f(a)0f(b) then there exists a c[a,b] such that f(c)=0.

Define the sequences (an) and (bn) inductively, as follows.


We note that

(bn-an)=2-n(b0-a0) (1)
f(an)0f(bn) (2)

By the fundamental axiom of analysisMathworldPlanetmath (an)α and (bn)β. But (bn-an)0 so α=β. By continuity of f


But we have f(α)0 and f(α)0 so that f(α)=0. Furthermore we have aαb, proving the assertion.

Set g(x)=f(x)-k where f(a)kf(b). g satisfies the same conditions as before, so there exists a c such that f(c)=k. Thus proving the more general result.

Title proof of intermediate value theorem
Canonical name ProofOfIntermediateValueTheorem
Date of creation 2013-03-22 12:33:56
Last modified on 2013-03-22 12:33:56
Owner yark (2760)
Last modified by yark (2760)
Numerical id 9
Author yark (2760)
Entry type Proof
Classification msc 26A06