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# Proof that Rn(0,1) is uncountable Watch

1. Hey, Just a bit of the proof that I don't understand so was hoping for someone to explain this to me.
To prove that the interval (0,1) is uncountable, we first assume its countable and list the elements as r_1, r_2, r_3,.... where r_i=0.a_i1a_i2....
We define b_i=1 if a_ii =/=1 and b_ii = 2 if a_ii=1. Then the real number r=0.b_1b_2... is not in the list and is in the interval (0,1).
Can someone please explain to me why r =/= r_i(i.e. why is r not in the list). Thanks!
Oh and sorry for the mess r_i is just r subscript i.
2. I presume you're working with decimal expansions. This is a simple diagonal argument; r isn't on the list simply because its decimal expansion differs from every number on the list somewhere.

(There are subtle issues with non-uniqueness of decimal expansions. A much better proof, in my opinion, uses the Cantor—Bernstein theorem to show that there is a bijection . This is a much stronger result, because it also shows precisely what the cardinality of the interval is.)
3. (Original post by Zhen Lin)
I presume you're working with decimal expansions. This is a simple diagonal argument; r isn't on the list simply because its decimal expansion differs from every number on the list somewhere.

(There are subtle issues with non-uniqueness of decimal expansions. A much better proof, in my opinion, uses the Cantor—Bernstein theorem to show that there is a bijection . This is a much stronger result, because it also shows precisely what the cardinality of the interval is.)
Thank you!

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Updated: December 27, 2010
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