Let us have a general sequence a in N^N
Question 1
If we simply insert that sequence into the nested fractions, then we get
a0+1/(a1+1/(a2+1/(a3+...)))
It seems that for any such sequence we get a converging sequence, which is a real number. So correct me if I am wrong, but that seems a straightforward bijection of all such sequences onto real numbers. Also, we are able to build a ordering simply in the following manner:
def compare(a,b): i = 0 while (a[i]==b[i]): if i%2 == 0: if a[i] b[i]: return "a < b" else: i += 1 else: if a[i] b[i]: return "a > b" else: i += 1
I never saw that construction alongside others like Dedekind, Weierstrass or Cantor approaches.
{Question 2}
Have anybody investigated the topologies on N^N for this mapping to be continous?
No, using Cantor's argument, once more. Consider all such sequences and now define a new one, that differs from the first one in the first position, from the second one in the second position and so on.
Well, actually, you are right , we have no natural bijection through the mechanism of continued fractions. We have the bijections as the following:
{NU{0}}^N contains all of the real nubmers which contians irrational numbers, which are bijective to N^N. So there shoud be a bijection, but not it that canonical way, since we can insert double zero elements in the sequence wherever we want.
It's not a question of double zeros. It's the property that all continued fractions are represented by the sequence of their coefficients. So, if we assume that this set is countable, the diagonal argument leads to a contradiction, since we can produce another element, that wasn't on the list, since the sequence of coefficients doesn't terminate. The fact that the sequence has integer elements doesn't mean it's countable.
Why should it be countable? N^N where N stands for Natural numers and has the continuum cardinality.
If it has the continuum cardinality, then Cantor's argument implies it's not countable.
Now the answer to the original question relies on proving that any real number can be represented as a continued fraction. These notes: https://www.math.uh.edu/~climenha/doc/ContFrac.pdf,
https://math.mit.edu/~rmd/IAP/continuedfractions.pdf
might be useful.(In particular cf. Theorem 9 of the second notes.)
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