Uniqueness of value of Riemann Integral(proof)

[Miike012]

The proof is in the document.

I highlighted the main points that I am questioning in the document.

I am questioning the fact that A = B...
(The following is in the document)
|A-B|=ε where they define the value of ε to be a positive arbitrary real number (ε>0).

And for A = B that means ε must be equal to zero however ε>0.

My Reasoning
For A≥B we have
A - B = ε
And for B≥A we have
A - B = -ε

And after solving the system of two eq. A - B = ε & A - B = -ε
we getA - B = 0
∴ A = B.
Is this how they proved A is the same as B?

 

[ArcanaNoir]

This is a standard move in analysis. If you want to show two things are equal, then you show that their difference is less than epsilon for an arbitrary epsilon. Since you can get the difference as close as you like to zero, it might as well be zero. Have you learned about things like supremum and infimum of a set?

 

[Miike012]

If you want to show two things are equal, then you show that their difference is less than epsilon for an arbitrary epsilon. Since you can get the difference as close as you like to zero, it might as well be zero.

Can you explain this mathematically? It doesn't make sense to me how you can say two things are equal just because they differ by an arbitrary value ε where we choose ε to → 0.
You can say two things are approximate though.

And I have not taken an analysis course.

 

[Miike012]

Deleted comment

 

[Office_Shredder]

|A-B|=ε where they define the value of ε to be a positive arbitrary real number (ε>0)

This is not what they have. If |A-B| = ε and ε was nonzero, then A and B would not be equal. What they have is |A-B|≤ ε.

 

[Miike012]

This is not what they have. If |A-B| = ε and ε was nonzero, then A and B would not be equal. What they have is |A-B|≤ ε.

Thank you. It makes sense now.

 

[HallsofIvy]

This is a standard move in analysis. If you want to show two things are equal, then you show that their difference is less than epsilon for an arbitrary epsilon. Since you can get the difference as close as you like to zero, it might as well be zero.0

I don't like the wording "it might as well be 0". That makes it sound like we can choose whether to have this number 0 or not. If, for any [itex]\epsilon> 0[/itex], [itex]z< \epsilon[/itex], z cannot be positive since then we could take [itex]\epsilon[/itex] (which can be any positive number) to be z itself and conclude that "z< z"/

Have you learned about things like supremum and infimum of a set?

 

[ArcanaNoir]

I don't like the wording "it might as well be 0". That makes it sound like we can choose whether to have this number 0 or not. If, for any [itex]\epsilon> 0[/itex], [itex]z< \epsilon[/itex], z cannot be positive since then we could take [itex]\epsilon[/itex] (which can be any positive number) to be z itself and conclude that "z< z"/

Good call. I concur.