interm.val.thm

# interm.val.thm - Intermediate-value Theorem Jonathan L.F...

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Unformatted text preview: Intermediate-value Theorem Jonathan L.F. King University of Florida, Gainesville FL 32611-2082, USA [email protected] Webpage http://www.math.ufl.edu/ ∼ squash/ 4 October, 2009 (at 14:00 ) Bernard Bolzano ( 1781–1848 ) proved the follow- ing form of the Intermediate-value Theorem . 1 : IVT. Suppose f : [ a,b ] → R is continuous, with f ( a ) and f ( b ) non-zero and having different signs. Then there exists a point c ∈ ( a,b ) which is a zero of f , i.e, f ( c ) = 0 . ♦ Proof. WLOGenerality, f ( a ) < 0 and f ( b ) > 0; otherwise, simply replace f by f ( which preserves continuity ) and note that a zero of f is a zero of f . Let L := a and R := b . For stage n = 1 , 2 ,... , either up to some integer K , or out to ∞ , I will produce numbers L n and R n such that: i[ n ] : a 6 L n- 1 6 L n < R n 6 R n- 1 6 b ; ii[ n ] : R n- L n = 1 2 [ R n- 1- L n- 1 ]; iii[ n ] : f ( L n ) < < f ( R n )....
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