Extreme Value Theorem Question (1 Viewer)

[VBN2470]

 

[seanieg89]

Hard to say how they want you to prove it without knowing exactly what you are supposed to know about real number / continuous functions already.

Do you know Bolzano-Weierstrass or Heine-Borel? Topological characterisations of continuity?

Two typical proofs:

1. Continuous functions preserve compact sets, so the range of f must be closed and bounded, and hence have a minimum, and this minimum must be positive since f is.

2. Assume that no positive minimum exists, so there is a sequence x_n with f(x_n)<1/n. Pass to a convergent subsequence by Bolzano-Weierstrass and this limit point a must satisfy f(a)=0...contradiction.

 

[Squar3root]

Hard to say how they want you to prove it without knowing exactly what you are supposed to know about real number / continuous functions already.

Do you know Bolzano-Weierstrass or Heine-Borel? Topological characterisations of continuity?

Two typical proofs:

1. Continuous functions preserve compact sets, so the range of f must be closed and bounded, and hence have a minimum, and this minimum must be positive since f is.

2. Assume that no positive minimum exists, so there is a sequence x_n with f(x_n)<1/n. Pass to a convergent subsequence by Bolzano-Weierstrass and this limit point a must satisfy f(a)=0...contradiction.

i think op is 1st year and that stuff you mentioned comes in 2nd or 3rd year. I'm assuming this is an [X] question (if you go to new south) so just skip it

 

[seanieg89]

i think op is 1st year and that stuff you mentioned comes in 2nd or 3rd year. I'm assuming this is an [X] question (if you go to new south) so just skip it

Well I am just curious, as I cannot imagine any proof that doesn't involve rigorous definition and study of the reals, which is generally beyond first year.

 

[seanieg89]

Unless you are literally allowed to just use the Extreme Value theorem without proof which makes it trivial.

(I initially assumed that the thread title referred to the fact that you are asked to prove something quite similar to the EVT.)

 

[VBN2470]

Unless you are literally allowed to just use the Extreme Value theorem without proof which makes it trivial.

(I initially assumed that the thread title referred to the fact that you are asked to prove something quite similar to the EVT.)

Yes, that is what they simply want us to do. I just wanted to know how you would do this question using the theorem (without any rigorous proof) hence the reason why I posted the question here..

 

[Carrotsticks]

Yes, that is what they simply want us to do. I just wanted to know how you would do this question using the theorem (without any rigorous proof) hence the reason why I posted the question here..

It's a bit strange because 'proving' the result via EVT is essentially just stating one half of the EVT with a wee bit of work.

 

[VBN2470]

It's a bit strange because 'proving' the result via EVT is essentially just stating one half of the EVT with a wee bit of work.

OK Thanks