I'd like to know if this proof of $\mathbb{R}$ being uncountable is correct. (It came to my mind while reading this proof.)
$\mathbb{R}$ is countable iff there is a surjective function $f:\mathbb{N} \mapsto \mathbb{R}$. In order for such a function to exist, it must be true that $\mathbb{N} \supseteq \mathbb{R}$. But that is false, which means $\mathbb{R}$ is not countable.