Completeness of R^2 with sup norm

[Fredrik]

From your post #43 you state

'Is it also intuitive that, we should use the properties of x_n and the sup norm to do that? Maybe not, but the problem tells you to do that.'

Those two sentences ended up kind of weird. It's not at all clear what I meant. I'll try to be more clear. The problem asked us to prove the following theorem:

The normed space $(\mathbb R^2,\|\ \|_\infty)$ is complete.​

(It also told us that you're allowed to use the theorem that says that $(\mathbb R,|\ |)$ is complete). Since a normed space is complete if and only if every Cauchy sequence in its underlying set is convergent, the theorem is equivalent to the following:

For all sequences $\langle x_n\rangle_{n=1}^\infty$ in $\mathbb R^2$, if $\langle x_n\rangle_{n=1}^\infty$ is Cauchy with respect to the sup norm, then $\langle x_n\rangle_{n=1}^\infty$ is convergent with respect to the sup norm.​

It should be obvious that it's impossible to prove this without using the definitions of "sup norm" and "Cauchy". OK, I suppose we could also do it by referring to some other theorem, which could be referring to yet another theorem, and so on, but the definitions of "sup norm" and "Cauchy" must both be used somewhere in the chain of proofs that ensure that all these theorems hold. If they're not used, we can't possibly claim that we have proved something about sequences that are Cauchy with respect to the sup norm.

Since our plan for the proof doesn't involve any references to other theorems about sequences that are Cauchy with respect to the sup norm, we definitely have to use the definitions of those terms in a step that derives something from the assumption that $\langle x_n\rangle_{n=1}^\infty$ is Cauchy with respect to the sup norm. Our step 1 is such a step. It proves that $\langle x_n\rangle_{n=1}^\infty$ (an arbitrary sequence in $\mathbb R^2$ that's Cauchy with respect to the sup norm) has the property that the sequences $\langle x_n(1)\rangle_{n=1}^\infty$ and $\langle x_n(2)\rangle_{n=1}^\infty$ are Cauchy with respect to the absolute value function on ℝ. So it wouldn't make any kind of sense to try to do step 1 without using the definitions of "sup norm" and "Cauchy".

That's how the theorem we're supposed to prove "tells you" to use the definitions of those terms in the first step. It tells you, simply by being a statement of the form "For all x, P(x) implies Q(x)", where P(x) is a statement about x that involves those terms. (In this case, P(x) is the statement that x is a sequence in ℝ² that's Cauchy with respect to the sup norm).

Since the problem explicitly tells us that we can use the theorem that says that $(\mathbb R,|\ |)$ is complete, I think it's very close to obvious that the person who wrote the problem intended us to solve it the way we did. The part of the problem statement I just mentioned is telling us to do steps 1 and 2, and step 3 is just to use what we found in steps 1-2 to complete the proof of the theorem.

 

[bugatti79]

Those two sentences ended up kind of weird. It's not at all clear what I meant. I'll try to be more clear. The problem asked us to prove the following theorem:

The normed space $(\mathbb R^2,\|\ \|_\infty)$ is complete.​

(It also told us that you're allowed to use the theorem that says that $(\mathbb R,|\ |)$ is complete). Since a normed space is complete if and only if every Cauchy sequence in its underlying set is convergent, the theorem is equivalent to the following:

For all sequences $\langle x_n\rangle_{n=1}^\infty$ in $\mathbb R^2$, if $\langle x_n\rangle_{n=1}^\infty$ is Cauchy with respect to the sup norm, then $\langle x_n\rangle_{n=1}^\infty$ is convergent with respect to the sup norm.​

It should be obvious that it's impossible to prove this without using the definitions of "sup norm" and "Cauchy". OK, I suppose we could also do it by referring to some other theorem, which could be referring to yet another theorem, and so on, but the definitions of "sup norm" and "Cauchy" must both be used somewhere in the chain of proofs that ensure that all these theorems hold. If they're not used, we can't possibly claim that we have proved something about sequences that are Cauchy with respect to the sup norm.

Since our plan for the proof doesn't involve any references to other theorems about sequences that are Cauchy with respect to the sup norm, we definitely have to use the definitions of those terms in a step that derives something from the assumption that $\langle x_n\rangle_{n=1}^\infty$ is Cauchy with respect to the sup norm. Our step 1 is such a step. It proves that $\langle x_n\rangle_{n=1}^\infty$ (an arbitrary sequence in $\mathbb R^2$ that's Cauchy with respect to the sup norm) has the property that the sequences $\langle x_n(1)\rangle_{n=1}^\infty$ and $\langle x_n(2)\rangle_{n=1}^\infty$ are Cauchy with respect to the absolute value function on ℝ. So it wouldn't make any kind of sense to try to do step 1 without using the definitions of "sup norm" and "Cauchy".

That's how the theorem we're supposed to prove "tells you" to use the definitions of those terms in the first step. It tells you, simply by being a statement of the form "For all x, P(x) implies Q(x)", where P(x) is a statement about x that involves those terms. (In this case, P(x) is the statement that x is a sequence in ℝ² that's Cauchy with respect to the sup norm).

Since the problem explicitly tells us that we can use the theorem that says that $(\mathbb R,|\ |)$ is complete, I think it's very close to obvious that the person who wrote the problem intended us to solve it the way we did. The part of the problem statement I just mentioned is telling us to do steps 1 and 2, and step 3 is just to use what we found in steps 1-2 to complete the proof of the theorem.

Fredrik, this is real good information. Thank you for taking the time out to write this.
It is really appreciated.