Math Is Fun Forum

cxc001

Member

Registered: 2010-04-09

Posts: 17

Use Extreme Value Theorem

Suppose f:[0,1]->R is continuous, f(0)>0, f(1)=0.
Prove that there is a X0 in (0,1] such that f(Xo)=0 & f(X) >0 for 0<=X<Xo (there is a smallest point in the interval [0,1] which f attains 0)

Since f is continuous, then there exist a sequence Xn converges to X0, and f(Xn) converges to f(Xo).
Since 0<=(Xo-1/n)<Xo
Can I just let Xn=Xo-1/n so that 0<=Xn<Xo
So when Xn->Xo, f(Xn)->f(Xo)

I wasn't convinced enough this is the right approach...

Alg Num Theory

Member

Registered: 2017-11-24

Posts: 693

Website

Re: Use Extreme Value Theorem

Last edited by Alg Num Theory (2019-05-20 01:57:35)

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Me, or the ugly man, whatever (3,3,6)