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==Proof of the non-countability of real numbers==

==Proof of the non-countability of real numbers==

There exists a map <math>f:[0,1]\rightarrow\mathbb{R}</math> (in fact all infinitly supported probability distribution does this). Therefor there are as many number in <math>[0,1]</math> as <math>\mathbb{R}</math>.

There exists a map <math>f:\mathbb{R}\rightarrow[0,1]</math> (in fact all infinitly supported probability distribution does this). Therefor there are as many number in <math>[0,1]</math> as <math>\mathbb{R}</math>.

We will now use [[proof by contradiction]] to show that the numbers in <math>[0,1]</math> are uncountable.

We will now use [[proof by contradiction]] to show that the numbers in <math>[0,1]</math> are uncountable.