Completeness of R^2 with sup norm

Fredrik

Bugatti79, you need to rethink your entire approach to proofs. Theorems are always implications, i.e. statements of the form ##A\Rightarrow B##, but you always ignore that. You always try to avoid making the assumption A, you always try to avoid using the definitions of the terms in A, and most of the time, you even assume B! These are the three biggest mistakes that can possibly be made in a proof. You also often make irrelevant assumptions that have nothing to do with the theorem.

You want to prove that if a sequence is Cauchy with respect to the ∞-norm, it's convergent with respect to the ∞-norm. So A is the statement "##\langle x_n\rangle## is Cauchy with respect to the ∞-norm", and B is the statement "##\langle x_n\rangle## is convergent with respect to the ∞-norm". And you start by assuming B, as usual. This is the single biggest mistake that can be made in a proof.

One thing you need to understand is that once a proof of ##A\Rightarrow B## has arrived at the statement B, there's nothing more to say. That's the end of the proof. So if you start by assuming B, nothing more needs to be said. In fact, it wouldn't make any sense to say anything more after that.

bugatti79

Let x_n=(x_n(1), x_n(2)) be an arbitrary Cauchy sequence in R^2. Need to show x_n is Cauchy in (R^2 || ||_\infy)
Claim: x_n(1) and x_n(2) are Cauchy sequences in R wrt || ||_\infty. To see this note

|x_n(1)-x_m(1)| <= sup|x_n(1)-x_m(1) = ||x_n(1)-x_m(1)||_\infty, similarly
|x_n(2)-x_m(2)| <= sup|x_n(2)-x_m(2) = ||x_n(2)-x_m(2)||_\infty

So x_n is Cauchy sequence in R which we know is complete (told this)
By definition of completeness, x_n \to x. Need to show x_n \to x in R^2, || ||_\infty

let ε>0 be given since x_n is Cauchy, there exists n_0 in N s.t. ||x_n-x_m||_\infty =sup|x_n-x_m|< ε for all n,m >= n_0 and x in R

let a=lim x for n \to infinity

by triangle inequality sup|x_n-x_m <= |x_n-a+a-x_m|<=|x_n-a|+|x_m-a|<=ε/2+ε/2<ε for all n,m >= n_0

implies x_n converges to x and therefore R^2 is complete wrt to sup norm...?

Fredrik

Now we're getting somewhere, but you keep making some pretty strange mistakes. You actually managed to get something wrong in every single statement you made. Somehow, this is still a decent attempt, because it looks like you have the right idea. You're just getting all of the details wrong. You have to be much more careful with the details. You're making it look like you're trying to get everything wrong. If you don't start making a much greater effort to get the details right, I think you will soon find it hard to get people to help you.

I have added some colored comments to your proof below.

Let x_n=(x_n(1), x_n(2)) be an arbitrary Cauchy sequence in R^2. You forgot to specify the norm with respect to which the sequence is Cauchy Need to show x_n is Cauchy convergent in (R^2 || ||_\infy)
Claim: x_n(1) and x_n(2) are Cauchy sequences in R wrt || ||_\infty That's not even a norm on ℝ. You should have said | |. To see this note

|x_n(1)-x_m(1)| <= sup|x_n(1)-x_m(1) sup_i |x_n(i)-x_m(i)| = ||x_n(1)-x_m(1)||_\infty This is wrong and doesn't make sense, since x_n(1) and x_m(1) are in ℝ and the ∞ norm is a norm on ℝ²., similarly
|x_n(2)-x_m(2)| <= sup|x_n(2)-x_m(2) = ||x_n(2)-x_m(2)||_\infty Same comments as for the preceding line

So x_n is Cauchy sequence in R which we know is complete (told this) x_n is a sequence in ℝ², not ℝ. But you probably meant x_n(1) and x_n(2).
By definition of completeness, x_n \to x. No, by definition of completeness, there exists an x(1) such that x_n(1)→x(1). Need to show x_n \to x in R^2, || ||_\infty That's not obvious at this point. What's obvious is that it's sufficient to show this, i.e. that if you succeed at showing this, you will be done.

let ε>0 be given since x_n is Cauchy, there exists n_0 in N s.t. ||x_n-x_m||_\infty =sup|x_n-x_m| sup_i|x_n(i)-x_m(i)| < ε for all n,m >= n_0 and x in R It's unclear if you meant "for all x in ℝ", or if you just meant to say that x is in ℝ. The former doesn't make sense, and the latter is wrong.

let a=lim x for n \to infinity You still haven't proved that x is convergent. But you probably meant something completely different from what you wrote

by triangle inequality sup|x_n-x_m <= |x_n-a+a-x_m|<=|x_n-a|+|x_m-a|<=ε/2+ε/2<ε for all n,m >= n_0

implies x_n converges to x and therefore R^2 is complete wrt to sup norm...? What are you doing here? Even the first thing you wrote down (sup|x_n-x_m) doesn't make sense.

Edit: I should perhaps have made it more clear that I think that you seem to have found the correct strategy for this proof. It seems that you understand the steps involved in the correct proof. It's just that you're making so many unnecessary mistakes when you try to write it down.

bugatti79

ok, another attempt..

Let x_n=(x_n(1), x_n(2)) be an arbitrary Cauchy sequence in (R^2, || ||_\infty). Need to show x_n is convergent in (R^2 || ||_\infy)
Claim: x_n(1) and x_n(2) are Cauchy sequences in (R, || ||). To see this note

||x_n-x_m||_\infty <= sup_i(|x_n(i)-x_m(i)|) = sup(|x_n(1)-x_m(1)|,|x_n(2)-x_m(2)|)

So x_n is Cauchy sequence in (R^2, || ||_\infty) which we know is complete (told this)
By definition of completeness, there exists an x_n(1) s.t x_n \to x.

Therefore it is sufficient to show x_n \to x in (R^2, || ||_\infty)

let ε>0 be given, since x_n is Cauchy there exists n_0 in N s.t. ||x_n-x_m||_\infty =sup_i|x_n(i)-x_m(i)|< ε/2 for all n,m >= n_0

Letting m \to \infty, we have |x_n(i)-x| < ε/2 for n >= n_0

ie ||x_n-x||_\infty < ε/2 < ε for n>=n_0

implies x_n \to x in (R^2, || ||_\infty)..?

Fredrik

Much better, but still not right.
There are three mistakes here.

1. The <= should actually be =, since that's just the definition of the sup norm. (I guess I should have mentioned that last time).
2. You didn't actually prove the claim you set out to prove, since you didn't write down the inequalities that prove that x_n(1) and x_n(2) are Cauchy.
3. After that incomplete attempt to prove the claim, you say "so", followed by the assumption you started with. This makes it look like you think you have just proved your starting assumption. When you start a sentence with "so", the next thing you say should be a consequence of the statements made before the "so". It shouldn't be a repeat of the starting assumption.

I didn't read the rest, since I had already found three mistakes.

bugatti79

Let x_n=(x_n(1), x_n(2)) be an arbitrary Cauchy sequence in (R^2, || ||_\infty). Need to show x_n is convergent in (R^2 || ||_\infy)
Claim: x_n(1) and x_n(2) are Cauchy sequences in (R, || ||).

Since we know x_n \to x \in R^2, || ||_\infty we know \exists n_0 \in N s.t

||x_n-x||_\infty < \epsilon \forall n \ge n_0 ie

||(x_n(1)-x(1), x_n(2)-x(2))||_\infty < \epsilon

We have that |x_n(1)-x(1)| \le max (|x_n(1)-x(1)|, |x_n(2)-x(2)|) < \epsilon
This shows x_n(1) \to x(1) as n \to \infty

Similarly

We have that |x_n(2)-x(2)| \le max (|x_n(1)-x(1)|, |x_n(2)-x(2)|) < \epsilon
This shows x_n(2) \to x(2) as n \to \infty

So x_n is Cauchy sequence in (R^2, || ||_\infty) which we know is complete (told this)

By definition of completeness, there exists an x_n(1) s.t x_n \to x.

||x_n-x_m||_\infty = sup_i(|x_n(i)-x_m(i)|) = sup(|x_n(1)-x_m(1)|,|x_n(2)-x_m(2)|)

Therefore it is sufficient to show x_n \to x in (R^2, || ||_\infty)

let ε>0 be given, since x_n is Cauchy there exists n_0 in N s.t. ||x_n-x_m||_\infty =sup_i|x_n(i)-x_m(i)|< ε/2 for all n,m >= n_0

Letting m \to \infty, we have |x_n(i)-x| < ε/2 for n >= n_0

ie ||x_n-x||_\infty < ε/2 < ε for n>=n_0

implies x_n \to x in (R^2, || ||_\infty)..?

Fredrik

The stuff in this quote is fine.

You can't make a statement that involves a variable "x" without first specifying its value, unless the statement is of the type "for all x..." or "there exists an x such that...". So there are three things I could guess that you meant to say here, but all of them suggest that we know that x_n is convergent with respect to the sup norm, and you don't know that. In fact, that's what we're trying to prove. Are you assuming what you're trying to prove again?

I had to stop there, since the mistake is so severe. I only had a quick look at the rest, and it seems like you have now abandoned the correct plan for this proof. Why? You were doing the right things last time. You were just doing them wrong.

I think it would be better if you don't color parts of your proof red. It just makes it harder to read.

Fredrik

OK, we're clearly not making much progress here. Since your problems with this are so extreme, I will describe the correct plan for this proof.

Assumption: x_n is Cauchy with respect to the sup norm.

Step 1: Prove that x_n(1) and x_n(2) are Cauchy with respect to the standard metric on R.

Step 2: Conclude that x_n(1) and x_n(2) are convergent with respect to the standard metric on R, and define x(1) and x(2) as the limits of those sequences.

Step 3: Guess that x_n converges to (x(1),x(2)), and prove that your guess is correct.

A few tips:

1. Don't ever state the assumption in a way that makes it look like you think it's something you have derived.

2. Don't ever assume the thing you're trying to prove.

3. Don't ever make a statement about a variable that you haven't defined, unless it's a "for all" or "there exists" statement.

4. Don't ever apply a function to something that's not in its domain.

5. Don't ever say that you're going to prove something and then try to prove something else.

6. Don't ever lose track of what your variables represent.

bugatti79

ok, I will get back to this asap. I need to cover other material in the mean time. Thanks

bugatti79

Step 1 and 2 only

Let x_n=(x_n(1), x_n(2)) be an arbitrary Cauchy sequence in (R^2, || ||_\infty). Need to show x_n is convergent in (R^2 || ||_\infy)

Claim: x_n(1) and x_n(2) are Cauchy sequences in (R, || ||).

Since x_n(1) and x_n(2) are Cauchy then there exist an epsilon>0 s.t.
||?||_\infty = sup |?| < ε for all ? >= ?

(The above line, I am not sure what to put in, confused with the lettering etc.
Assuming I can get the above correct I think I can continue and safely say...)

Since it is given that R is complete, ie the real line is complete, then we can say x_n(1) and x_n(2) converge to x(1) and x(2) respectively...?

Fredrik

That last thing should be (R, | |), since the norm on ℝ is just the absolute value.

No. Since they are Cauchy, for all ε>0, there exists a positive integer N such that...

The sup norm (which is a norm on ℝ²) has nothing to do with this part, since we're talking about Cauchy sequences in ℝ. What you need to do here is to just use the definition of "Cauchy sequence" to figure out how to end the "for all" statement I started above. You just need to understand what the statements "x_n(1) is a Cauchy sequence" and "x_n(2) is a Cauchy sequence" mean.


You can't say that ##x_n\to x(1)## until you have defined what x(1) is.

bugatti79

Since they are Cauchy, for all ε>0, there exists a positive integer N such that

|x_m-x_n|< epsilon for m,n >= N..?

Fredrik

The expression |x_n-x_m| doesn't really make sense, does it? For each n, x_n is in ℝ², but the absolute value is a norm on ℝ. I sugggest you take another look at my list of tips in post #25, in particular tip number 4.

Also, the statement you're trying to use is about x_n(1) and x_n(2). You can't immediately jump to a conclusion about x_n.

bugatti79

Since they are Cauchy, for all ε>0, there exists a positive integer N such that

|x_a-x_n(1)|< epsilon for a,n >= N and similarly

|x_b-x_n(2)|< epsilon for b,n >= N

or

Since they are Cauchy, for all ε>0, there exists a positive integer N_(1,2) such that

|x_m-x_n(1)|< epsilon for m,n >= N_1 and similarly

|x_m-x_n(2)|< epsilon for m,n >= N_2 ..?

Fredrik

Yes, both of those are correct. The first one is the one we want. Note that it follows from the second one. (We can define N=max{N_1,N_2}).

Instead of "there exists a positive integer N_(1,2)", it would have been better to say "there exist positive integers N_1 and N_2".

Not really important, but perhaps still worth mentioning: If you need more than two symbols for integers, and you have already used n and m, I think you should use i,j or k. This is more of a tradition than anything else.

bugatti79

Since they are Cauchy, for all ε>0, there exists positive integers N_1 and N_2 such that

|x_m-x_n(1)|< epsilon for m,n >= N_1 and similarly

|x_m-x_n(2)|< epsilon for m,n >= N_2. We can define N=(N_1, N_2)

so |x_m-x_n(1)| and |x_m-x_n(2)| both tend to 0 when m,n tend to infinity.

Hence x_n(1) and x_n(2) are Cauchy, but since we know R is complete, these Cauchy's must converge, therefore let x_(1) and x_(2) be the limits of x_n(1) and x_n(2) respectively...?

Fredrik

Oh, wait, when I said those two were correct, I didn't see that you had written things like |x_m-x_n(1)|. This clearly doesn't make sense. x_m is in ℝ², and x_n(1) is in ℝ, so the difference x_m-x_n(1) is nonsense. I thought you had written |x_m(1)-x_n(1)|.

There are several other (enormous) problems here. I may have caused some of your confusion, because I sort of forgot what we were doing for a while, and only looked at whether your statements made sense individually. You were supposed to do what I called "step 1" in post #25, i.e. you were supposed to prove that prove that x_n(1) and x_n(2) are Cauchy with respect to the standard metric on R. Since I had temporarily forgotten that, I didn't even realize that you opened with "Since x_n(1) and x_n(2) are Cauchy...".

I don't even know what to say. How many times have you been told that you can't assume what you're trying to prove? Is this concept hard to understand? Consider this "proof" of the claim that the moon is made of cheese:

Since the moon is made of cheese, [insert anything you want here]. Therefore, the moon is made of cheese.

Don't you see how absurd this is? If you assume the thing you're trying to prove, you can prove anything. You can prove that every statement is true, even the ones that are known to be false. You need to continue to think about this until you are absolutely sure that you understand it perfectly.