Real Numbers are Uncountably Infinite

Theorem
The set of real numbers $\R$ is uncountably infinite.

Historical Note
This was first demonstrated by.

Cantor's first and second proofs given above are less well known than the diagonal argument, and were in fact downplayed by Cantor himself: the first proof was given as an aside in his paper proving the countability of the algebraic numbers.

conjectures that this was because Cantor feared opposition from among other contemporaries who aggressively dismissed Cantor's ideas.

In particular Cantor's first proof is worth reading; several texts reject the first proof as being more complicated and less instructive, but this seems to have arisen because the Diagonal argument has proven to be a more versatile tool and thus the others forgotten and dismissed.

In this instance the first and second proofs of Cantor are of equal merit to the diagonal argument, and the three together present clearly the flavor of the theorem.