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
?)
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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??