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??