Corollary 7.4.4
##### Claim

The set $\mathbb{R}$ of real numbers is uncountable.

Proof
For sake of contradiction, we will assume that $\mathbb{R}$ is countable. By the definition of countable there exists a 1-1 correspondence $f:\mathbb{N}\to\mathbb{R}$. In order to show our contradiction we must construct a new function:

$f*:X\to(1,0)$ via
$f*(n)=f(n)$

By assuming $f*$ to be onto and one to one we can see that $cardx=card(0,1)$. Therefore $(0,1)$ is countable, however, by theorem 7.4.3, $(0,1)$ is uncountable. So by contradiction $\mathbb{R}$ is uncountable.

Proof

Since $(0,1)$ is a subset of $\mathbb{R}$ and $(0,1)$ is uncountable by theorem 7.4.3, $\mathbb{R}$ must be uncountable.
$\blacksquare$
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