So, I was recently working on a project exploring the Cauchy sequence approach to constructing the real numbers. As I was working through the proofs that multiplication, addition and multiplicative inversion of the reals were well defined within this structure I noticed that the proofs that each operation preserved the Cauchy criterion were basically exact mirrors of the proofs showing that they also preserved equivalence...So here's my question.
Let f be a function defined on the reals. Let F be a function defined on the set of all sequences of reals as follows:
[math]F(\left\{a_n\right\})=\left\{f(a_n)\right\}[/math]
And if * is an arbitrary binary operation on the reals... let us define a binary operation on sequences of reals as follows:
[math]\left\{a_n\right\}\cdot\left\{b_n\right\}=\left\{a_n*b_n\right\}[/math]
Then are the following two statements equivalent?
F maps all Cauchy sequences to Cauchy sequences
F if two Cauchy sequences are equivalent (their difference converges to zero) then their images are equivalent (note that this merely means the difference approaches zero..not necessarily that they are Cauchy)
And are these statements also equivalent?
* maps any pair of Cauchy sequences to another Cauchy sequence.
If {a} and {b} are equivalent and {c} and {d} are equivalent, then {a*c} is equivalent to {c*d}
Question about Cauchy Sequences
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Question about Cauchy Sequences
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Re: Question about Cauchy Sequences
Dark Avorian wrote:Let f be a function defined on the reals. Let F be a function defined on the set of all sequences of reals as follows:
[math]F(\left\{a_n\right\})=\left\{f(a_n)\right\}[/math]
And if * is an arbitrary binary operation on the reals... let us define a binary operation on sequences of reals as follows:
[math]\left\{a_n\right\}\cdot\left\{b_n\right\}=\left\{a_n*b_n\right\}[/math]
Then are the following two statements equivalent?
F maps all Cauchy sequences to Cauchy sequences
F if two Cauchy sequences are equivalent (their difference converges to zero) then their images are equivalent (note that this merely means the difference approaches zero..not necessarily that they are Cauchy)
Yes. Suppose F maps Cauchy sequences to Cauchy sequences. Let A = {a_{n}} and B = {b_{n}} be equivalent Cauchy sequences. If F(A) and F(B) were not equivalent, then the Cauchy sequence given by interleaving terms C = a_{1}b_{1}a_{2}b_{2}… would have an image that is not Cauchy. This proves the forward implication.
Suppose instead that F sends equivalent Cauchy sequences to equivalent sequences. Let A = {a_{n}} be any Cauchy sequence, and let B = {b_{n} = b} be the unique constant sequence equivalent to A, which must exist by completeness of the reals. Clearly F(B) is Cauchy, since it is a constant sequence with every term equal to f(b). But F(A) is equivalent to F(B), so it must also be Cauchy. This proves the reverse implication, hence equivalence.
Dark Avorian wrote:And are these statements also equivalent?
* maps any pair of Cauchy sequences to another Cauchy sequence.
If {a} and {b} are equivalent and {c} and {d} are equivalent, then {a*c} is equivalent to {c*d}
Yes. Look at Cauchy sequences (A, B) in ℝ^{2}, whence * is just a function f:ℝ^{2}→ℝ. Now the previous result applies.
wee free kings
Re: Question about Cauchy Sequences
I was thinking about this some more, wondering if it holds for incomplete spaces. The forward implication clearly holds, as my proof uses only properties of Cauchy sequences. But in proving the reverse implication, “If F sends equivalent Cauchy sequences to equivalent sequences, then F sends Cauchy sequences to Cauchy sequences”, I used completeness.
My intuition says it should be true even for incomplete spaces, because every metric space is a subset of its formal completion by Cauchy sequences, and the statement has to be true in that superspace. Does anyone have a concise proof that stays within the original incomplete space? (Or a counterexample?)
My intuition says it should be true even for incomplete spaces, because every metric space is a subset of its formal completion by Cauchy sequences, and the statement has to be true in that superspace. Does anyone have a concise proof that stays within the original incomplete space? (Or a counterexample?)
wee free kings
Re: Question about Cauchy Sequences
Suppose {x_{n}} is Cauchy, but {f(x_{n})} is not Cauchy. Then we can find some ε > 0 and an increasing sequence {n_{k}} such that d(f(x_{n_k}), f(x_{n_{k+1}})) > ε for all k. Then {x_{n_k}} and {x_{n_{k+1}}} are equivalent Cauchy sequences since {x_{n}} is Cauchy, but their images {f(x_{n_k})} and {f(x_{n_{k+1}})} are not equivalent sequences.
This should prove the reverse implication without assuming completeness, assuming I didn't make any dumb mistakes at this late hour.
This should prove the reverse implication without assuming completeness, assuming I didn't make any dumb mistakes at this late hour.
Re: Question about Cauchy Sequences
Nitrodon wrote:Suppose {x_{n}} is Cauchy, but {f(x_{n})} is not Cauchy. Then we can find some ε > 0 and an increasing sequence {n_{k}} such that d(f(x_{n_k}), f(x_{n_{k+1}})) > ε for all k. Then {x_{n_k}} and {x_{n_{k+1}}} are equivalent Cauchy sequences since {x_{n}} is Cauchy, but their images {f(x_{n_k})} and {f(x_{n_{k+1}})} are not equivalent sequences.
This should prove the reverse implication without assuming completeness, assuming I didn't make any dumb mistakes at this late hour.
Nicely done, thanks.
wee free kings
Re: Question about Cauchy Sequences
Nitrodon wrote:Then we can find some ε > 0 and an increasing sequence {n_{k}} such that d(f(x_{n_k}), f(x_{n_{k+1}})) > ε for all k.
Is this obvious? I'm prone to missing simple things at the best of times and I've not done any proper maths in a while, so bear with me. I know that directly from the definition of notCauchy we get two increasing sequences {n_{k}} and {m_{k}} such that d(f(n_{k}), f(m_{k})) > ε for all k, but I'm pretty sure you can't stitch these together in an obvious way (in that it's not too hard to come up with two sequences satisfying this condition for a given ε but where all other pairwise distances are < ε). Obviously this is academic since your proof works with the sequences {n_{k}} and {m_{k}}, but I'd be interested to know...
All posts are works in progress. If I posted something within the last hour, chances are I'm still editing it.
Re: Question about Cauchy Sequences
You're right that you can easily make the other pairwise distances < ε. However, there will always be enough pairwise distances > ε/2 by the triangle inequality, so it's possible to "stitch together" the sequences by using ε/2 instead of ε.
I somehow didn't think of just using {n_k} and {m_k}, which would have saved me the trouble of having to determine that fact.
I somehow didn't think of just using {n_k} and {m_k}, which would have saved me the trouble of having to determine that fact.
Re: Question about Cauchy Sequences
Ah, of course  if you make sure that you always have n_{k} < m_{k} < n_{k+1}, you can start at n_{1} and inductively from n_{k} either go straight to n_{k+1} or go via m_{k}. I was trying to make the triangle inequality work for me, but didn't think of applying pigeonhole >.<
All posts are works in progress. If I posted something within the last hour, chances are I'm still editing it.
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