See >>here<< (you will need to fill in a gap of two yourself)
CB
Hey guys, I have a problem and I don't know a good way to deal with it... so basically, it says that a number is called Liouville number, for every there exist integers p and q > 1 such that
question: prove that the set of all Liouville numbers has the same cardinality as the
Please, how can I approach this question??? is there a way to construct a bijection from the Liouvilles to the reals???
OK, I see the answer. Not hard at all. Thanks a lot
Hmmm... just one doubt, for the proof to work you need to assume a n to be large enough. How one can justified that?
Finally, we can prove that x is irrational, but do we need to prove that the set of all Liouville numbers is actually equal to the set of the irrationals?
I read that as for any n>0 there are a p and q with the required properties? (the paragraph on Liouville's constant can be adapted to do that)
There is a subset of the Liouville numbers that is equinumerous with the reals (those zero except at the n! decimal place), and that should be sufficient to prove that the Liouville numbers are equinumerous with the reals.Finally, we can prove that x is irrational, but do we need to prove that the set of all Liouville numbers is actually equal to the set of the irrationals?