I have a problem on this question.
Suppose and for every list of finitely many distinct elements of . Prove that is countable.
(Hint: For each positive integer , let . What can you say about the number of elements in ?)
I can only conclude from a hint that (where = the number of elements in ). It seems there is no link from this hint to the question. I need to show either there is a function that is one-to-one, or a function that is onto.
If A = (0, 1), clearly this is uncountable. But I may choose b to be very large so that for any . Then, how could that A needs to be countable??
Thank you very much in advance.