Orchid XP v8 wrote:
> Darren New wrote:
> 
>> Map each real number to a positive integer.
> 
> Um... like, how? There's more of them...

It's a proof by contradiction, so you start by assuming that you can 
build such a map and then derive from that a contradiction, thus 
invalidating the premise that you the map was possible.

It's wise to learn not to trust your intuitions too much with this sort 
of thing.  As warp pointed out, there are the same number of rationals 
as natural numbers, whereas intuitively you might think that there's 
more rationals than integers and the same number of reals as rationals. 
And intuition becomes even less of a guide when you start reasoning 
about more complicated sets.

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