Real Numbers are Uncountably Infinite/Cantor's Diagonal Argument

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

Cantor's Diagonal Argument
We show that the unit interval $[0. . 1]$ is uncountable, from which uncountability of $\R$ follows immediately.

Suppose that $[0. . 1]$ is countable.

Clearly $[0. . 1]$ is not finite because $\displaystyle \frac 1 n$ are distinct for $n \in \N$.

Therefore an injection $[0. . 1]\hookrightarrow \N$ enumerates $[0. . 1]$ with a subset of the natural numbers.

By relabeling, we can associate each $x \in [0. . 1]$ to precisely one natural number to obtain a bijection.

Let $g$ be such a correspondence:

where juxtaposition of digits descibes the decimal expansion of a number.

Note that the decimal expansion is not directly unique, because of 0.999...=1.

This problem is overcome by disallowing infinite strings of $9$'s in the decimal expansion.

For every $k \in \N$ define $f_k = d_{kk} + 1$ taken modulo $10$.

Let $y$ be defined by the decimal expansion:


 * $y = 0.f_1 f_2 f_3 \ldots$

Now


 * $y$ differs from $g(1)$ in the first digit of the decimal expansion
 * $y$ differs from $g(2)$ in the second digit of the decimal expansion

and generally the $n^\text{th}$ digit of the decimal expansion of $g(n)$ and $y$ is different.

By Existence of Base-N Representation, any decimal expansion of a real number is unique (except for the issue from 0.999...=1).

So $y$ can be none of the numbers $g(n)$ for $n \in \N$.

But $g$ is a bijection, a contradiction.

Historical Note
This proof was first demonstrated by Georg Cantor.