Prove Integral of f in [0,1] with Homework Statement

[daniel_i_l]

Homework Statement

f is a function defined in [0,1] by:
If there's some n (natural number) so that x = 1/n then f(x) = 1
If not then f(x) = 0.
Prove or disprove:
1) For every E>0 there's some P (P is a division of [0,1], for example
{0,0.2,0.6,1}) so that S(P) < E (Where S(P) is the upper some of f(x) with the division P)

2)f in integrable in [0,1] and the integral is 0.

Homework Equations

The Attempt at a Solution

1) True, because we can always choose A<E and: P={0,A,1/n,...,1/2,1} were 1/n is the smallest number of that type bigger than A. And then
S(P) =A < E

2)True, because if for every E>0 there's someP so that S(P) < E than the upper integral of f in [0,1] is 0. And since the lower integral is always bogger or equal to 0 (in this case) but smaller than the upper integral then the upper and lower integrals are both equal to 0 and so the integral exists and is 0.

Are thise right?
Thanks.

 

[Dick]

{0,0.2,0.6,1} is not a division of [0,1]. It's a set of points in [0,1]. How do you mean this to be interpreted as a 'division'? What you want is to choose the width of an interval around 1/n (say r_n) in such a way that the sum of the r_n is less than E. There are many explicit ways to do this. Choose one.

 

[daniel_i_l]

When I wrote {0,0.2,0.6,1} I meant {[0,0.2].[0.2,0.6],[0.6,1]}.

 

[Dick]

When I wrote {0,0.2,0.6,1} I meant {[0,0.2].[0.2,0.6],[0.6,1]}.

Ok. The previous advice still stands. Put 'small intervals' around each of the points 1/n.

 

[daniel_i_l]

Thanks, and after I do what you said to prove that (1) is true, then does that also mean that (2) is true (as I explained in the answer to 2)?

 

[Dick]

Thanks, and after I do what you said to prove that (1) is true, then does that also mean that (2) is true (as I explained in the answer to 2)?

It sure does. You then have that the upper and lower sum are both zero. So what's your choice for a unit interval division. You can be pretty explicit.

 

[Ryker]

Ok. The previous advice still stands. Put 'small intervals' around each of the points 1/n.

Sorry to resurrect an old thread, but I have the same problem as the OP's second one, and haven't yet found a good solution for it yet. But I don't think what you suggested here is correct, since the set containing 1/n is infinite, whereas the partitions need to be finite sets. So you can't really put small intervals around each 1/n. It would be great, if you could, and that was my first thought when approaching the problem, as well, but I don't think you can.

Any other suggestions perhaps?

 

[Dick]

Sorry to resurrect an old thread, but I have the same problem as the OP's second one, and haven't yet found a good solution for it yet. But I don't think what you suggested here is correct, since the set containing 1/n is infinite, whereas the partitions need to be finite sets. So you can't really put small intervals around each 1/n. It would be great, if you could, and that was my first thought when approaching the problem, as well, but I don't think you can.

Any other suggestions perhaps?

Nope. Same suggestion. Try an interval size like 1/2^(n+k) and then let k->infinity to shrink the partition. If you need a finite partition then you just have to be a little fussier. Cover the first N points with intervals of those sizes and then cover the remaining points with one of size 1/N. Now let N->infinity as well.

 

[Ryker]

Nope. Same suggestion. Try an interval size like 1/2^(n+k) and then let k->infinity to shrink the partition. If you need a finite partition then you just have to be a little fussier. Cover the first N points with intervals of those sizes and then cover the remaining points with one of size 1/N. Now let N->infinity as well.

Well, it's not me who needs a finite partition, it's the definition

Could you perhaps elaborate on your suggestion? When you say cover the first N points with intervals of those sizes, what size do you mean? Because I can't fix n (it's infinite), I can only fix k. And if I introduce N, as well, then I now have two variables, that are not dependent on one another. But then I can't take the limit, I can only let them approach infinity one at a time.

 

[Dick]

Well, it's not me who needs a finite partition, it's the definition

Could you perhaps elaborate on your suggestion? When you say cover the first N points with intervals of those sizes, what size do you mean? Because I can't fix n (it's infinite), I can only fix k. And if I introduce N, as well, then I now have two variables, that are not dependent on one another. But then I can't take the limit, I can only let them approach infinity one at a time.

Ok, put k=N. Surround the point 1/n with an interval of size 1/2^(n+N) for n from 1 to N. That's finite, yes? The last point is 1/N. So you can include the rest of the points in an interval of size 1/N. That's one more interval. Still finite. Now take N to infinity.

 

[Ryker]

Now you're talking Thanks, I think this should work now. I had a similar idea, but couldn't quite wrap my head around how to put it formally.