Why the B-W Theorem is used when proving continuity implies uniform continuity?

[schniefen]

Homework Statement

Using the Bolzano-Weierstrass Theorem (BWT) to prove continuity implies uniform continuity.

Relevant Equations

Definitions of continuity, uniform continuity and BWT á la Wikipedia.

In my textbook when proving continuity implies uniform continuity (which is very similar to the proof given here), BWT is used to find a converging subsequence. I cannot see why this is needed. Referring to the linked proof, if we open up the inequality $|x_n-y_n|<\frac{1}{n}$, isn't by the squeeze theorem then $(x_n-y_n)=0\iff x_n=y_n$, and so one can conclude these sequences converge to the same limit and jump the BWT step in the proof.

 

[fresh_42]

Squeezing is how Bolzano Weierstraß is proven, so you can of course substitute one by the other.

 

[schniefen]

Squeezing is how Bolzano Weierstraß is proven, so you can of course substitute one by the other.

So it would be correct to jump the BWT step and instead apply the squeeze theorem directly to $(x_n-y_n)$? Why would one choose to use the BWT in the first place?

 

[fresh_42]

It is shorter to say. You also need topological completeness or closure for your argument, and prove why it can be applied. So you will mimic the proof of Bolzano-Weierstraß, just mention the theorem is shorter.

 

[WWGD]

Is the official statement that continuity in a compact set implies uniform continuity?

 

[schniefen]

Yes

 

[WWGD]

Problem Statement: Using the Bolzano-Weierstrass Theorem (BWT) to prove continuity implies uniform continuity.
Relevant Equations: Definitions of continuity, uniform continuity and BWT á la Wikipedia.

In my textbook when proving continuity implies uniform continuity (which is very similar to the proof given here), BWT is used to find a converging subsequence. I cannot see why this is needed. Referring to the linked proof, if we open up the inequality $|x_n-y_n|<\frac{1}{n}$, isn't by the squeeze theorem then $(x_n-y_n)=0\iff x_n=y_n$, and so one can conclude these sequences converge to the same limit and jump the BWT step in the proof.

The issue, I don't see if you addressed this, is that a fixed value of n will work for all points, instead of having n vary for different points. EDIT: I don't see how the squeeze theorem warrants this . I think you should be more precise. Do you mean $|x_n -y_n| < 1/n \forall n$?

 

[schniefen]

The issue, I don't see if you addressed this, is that a fixed value of n will work for all points, instead of having n vary for different points.

How does that connect to the use of the Bolzano-Weierstrass theorem?

 

[WWGD]

How does that connect to the use of the Bolzano-Weierstrass theorem?

How does that connect to the use of the Bolzano-Weierstrass theorem?

I was referring more to the uniform continuity part.

 

[WWGD]

It seems you can choose a value $\epsilon$ construct at each point a convergent subsequence , choose a cover from which you extract a finite subcover... to find a value of n so that 1/n will work for all.

 

[schniefen]

I’m not sure I understand. I’ve been told that the application of the Bolzano Weierstrass theorem is necessary “to have a sequence to take the limit of”. It confuses me, because the proof already assumes we have two sequences $x_n$ and $y_n$ that one could perfectly take the limit of, no?

 

[WWGD]

Take a sequence about each point. By BW there is a convergent subsequence. This can done to have a sequence convergent to any point. Since sequence converges, fix an $\epsilon$, so you can by convergence, find an $N_{\epsilon}$ so that $|x_n -y_n | < \epsilon$ $\forall n>N$ . Now the collection of different values $N_{\epsilon}$ will have a minimum, which will give you the uniform continuity.