Can anybody explain this to me? (Analysis)

[Artusartos]

For question 32.2 in this link:

http://people.ischool.berkeley.edu/~johnsonb/Welcome_files/104/104hw11sum06.pdf

I did not understand how $b^2/2 \leq U(f)$. We know that we have strict inequality in $t_{k+1} > \frac{t_k + t_{k+1}}{2}$...so don't we need to have $b^2/2 < U(f)$ instead of $b^2/2 \leq U(f)$?

Thanks in advance

 

[Ray Vickson]

For question 32.2 in this link:

http://people.ischool.berkeley.edu/~johnsonb/Welcome_files/104/104hw11sum06.pdf

I did not understand how $b^2/2 \leq U(f)$. We know that we have strict inequality in $t_{k+1} > \frac{t_k + t_{k+1}}{2}$...so don't we need to have $b^2/2 < U(f)$ instead of $b^2/2 \leq U(f)$?

Thanks in advance

If a strict inequality folds for any finite partitions, all you can conclude is that the non-strict inequality holds in the limit, or when taking the sup, etc. In other words, we cannot have U(f) < b^2/2, because in order to have that we would have to have U_P(f) < b^2/2 for some finite partition P.

To put it another way: we have 1/n > 0 for all positive integers n, but inf{1/n: n is a positive integer} = 0.

RGV

 

[Artusartos]

If a strict inequality folds for any finite partitions, all you can conclude is that the non-strict inequality holds in the limit, or when taking the sup, etc. In other words, we cannot have U(f) < b^2/2, because in order to have that we would have to have U_P(f) < b^2/2 for some finite partition P.

To put it another way: we have 1/n > 0 for all positive integers n, but inf{1/n: n is a positive integer} = 0.

RGV

Thanks, but...

"we cannot have U(f) < b^2/2, because in order to have that we would have to have U_P(f) < b^2/2 for some finite partition P."...but if you look at the last part of the solution, it actually shows that U(f,P) < b^2/2 for all partitions P...

 

[Ray Vickson]

Thanks, but...

"we cannot have U(f) < b^2/2, because in order to have that we would have to have U_P(f) < b^2/2 for some finite partition P."...but if you look at the last part of the solution, it actually shows that U(f,P) < b^2/2 for all partitions P...

OK, so I got the inequalities reversed. The argument is still the same.

RGV

 

[Artusartos]

OK, so I got the inequalities reversed. The argument is still the same.

RGV

No I wasn't saying that you got the inequalities reversed...

I was just saying that (at the very end of the solution), it is shown that U(f,P)> b^2/2 for all P.

 

[haruspex]

OK, so I got the inequalities reversed.

No, you had it right the first time.
Artusartos, Ray is saying the argument runs like this.
We know that for any finite partition U_P(f) > b²/2. Suppose in the limit U(f) < b²/2. That would imply U_P(f) < b²/2 for some finite partition P, contradicting what we know. Therefore U(f) ≥ b²/2.
The OP says, in part:

don't we need to have b²/2<U(f) ?​

Therein lies a misunderstanding. The relationship between b²/2 and U(f) is not a condition that's needed, it's a relationship we are trying to deduce. The point is that we cannot deduce U(f) > b²/2. This is because it is the limit of a sequence, and a sequence in which every term is > x can be equal to x in the limit. Instead, we can deduce the weaker relationship U(f) ≥ b²/2.

 

[Artusartos]

No, you had it right the first time.
Artusartos, Ray is saying the argument runs like this.
We know that for any finite partition U_P(f) > b²/2. Suppose in the limit U(f) < b²/2. That would imply U_P(f) < b²/2 for some finite partition P, contradicting what we know. Therefore U(f) ≥ b²/2.
The OP says, in part:

don't we need to have b²/2<U(f) ?​

Therein lies a misunderstanding. The relationship between b²/2 and U(f) is not a condition that's needed, it's a relationship we are trying to deduce. The point is that we cannot deduce U(f) > b²/2. This is because it is the limit of a sequence, and a sequence in which every term is > x can be equal to x in the limit. Instead, we can deduce the weaker relationship U(f) ≥ b²/2.

Thanks a lot, I think I understand it more now...but...

The other poster says "In other words, we cannot have U(f) < b^2/2, because in order to have that we would have to have U_P(f) < b^2/2 for some finite partition P." So (according to this statement) if we know that U_P(f) < b^2/2 for "some finite" partition P, then we can say that U(f) < b^2/2. In the text that I attached, it says that U_P(f) < b^2/2 for "all" finite partitions P...so why can't we say the same thing? Since, if we have U(f)<b^2/2 for "all" partitions, then we definitely have it for "some" partitions...

 

[haruspex]

"In other words, we cannot have U(f) < b^2/2, because in order to have that we would have to have U_P(f) < b^2/2 for some finite partition P." So (according to this statement) if we know that U_P(f) < b^2/2 for "some finite" partition P, then we can say that U(f) < b^2/2.

No, that doesn't follow. You're turning A $\Rightarrow$ B into B $\Rightarrow$ A:

If {U(f) < b^2/2} then {$\exists$ P s.t. U_P(f) < b^2/2}​

is not the same as

If {$\exists$ P s.t. U_P(f) < b^2/2} then {U(f) < b^2/2}​

 

[Artusartos]

No, that doesn't follow. You're turning A $\Rightarrow$ B into B $\Rightarrow$ A:

If {U(f) < b^2/2} then {$\exists$ P s.t. U_P(f) < b^2/2}​

is not the same as

If {$\exists$ P s.t. U_P(f) < b^2/2} then {U(f) < b^2/2}​

Oh, ok...I get it. Thanks :)