Theorem: The set of finite sequences of elements of a countable set Image may be NSFW.
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is countable.
I like this result because it specializes to several other basic countability results: for example, it implies that countable unions, finite products, and the set of finite subsets of countable sets are countable. I know several proofs of this result and I am honestly curious which ones people prefer.
Proof 1
Map a finite sequence Image may be NSFW.
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of positive integers to the positive integer Image may be NSFW.
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. This map is a bijection by the uniqueness of binary expansion.
Proof 2
Map a finite sequence Image may be NSFW.
Clik here to view. of non-negative integers to the positive integer Image may be NSFW.
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. This map is a bijection by the uniqueness of prime factorization.
Proof 3
There are finitely many sequences Image may be NSFW.
Clik here to view. of positive integers with Image may be NSFW.
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for a fixed positive integer Image may be NSFW.
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and a countable union of finite sets is countable. In the interest of being more explicit about the bijection here, one can order these sequences first by Image may be NSFW.
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and then lexicographically.
Proof 4
If you believe that Image may be NSFW.
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is countable, map a finite sequence Image may be NSFW.
Clik here to view. of positive integers to the continued fraction Image may be NSFW.
Clik here to view.![[a_1; a_2, a_3, ... a_n + 1]](http://s0.wp.com/latex.php?latex=%5Ba_1%3B+a_2%2C+a_3%2C+...+a_n+%2B+1%5D&bg=ffffff&fg=333333&s=0&c=20201002)
. This map is a bijection by the uniqueness of continued fraction expansions (as long as they end in a positive integer greater than Image may be NSFW.
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).
Composing this map with the inverse of the map in Proof 1 gives an explicit enumeration of the rationals. I believe it was on the Putnam one year.
Question
Does anyone know any other genuinely different proofs?
