Question

Real numbers can be represented in the decimal expansion form $r_1=n+0.d_1d_2d_3\dots d_\infty$, where $n$ is an integer, and $d_i$ is a digit between 0 and 9.
Real numbers can also be represented the binary expansion form $r_2=n+0.b_1b_2b_3\dots b_\infty$, where $n$ is an integer, and $b_i$ is a digit either 0 or 1.
Show that the set of all possible $r_1$ is the same size as the set of all possible $r_2$.
You may use the identity $k\times\infty=\infty$.

Solution

Each $d_i$ can take 10 possible values, therefore the decimal part of a real number $d_1d_2d_3\dots d_\infty$ can take $10^\infty$ possible values. In total $r_1$ can take $\infty\times 10^\infty$ possible values.
Each $b_i$ can take 2 possible values, therefore the decimal part of a real number $d_1d_2d_3\dots d_\infty$ can take $2^\infty$ possible values. In total $r_1$ can take $\infty\times 2^\infty$ possible values.

We need to show that $\infty\times 10^\infty$ is the same as $\infty\times 2^\infty$.

$$ \begin{align*} 10^\infty&=(2^{\log_2 10})^\infty\\ &=2^{\log_2 10\times\infty} \end{align*} $$

By the identity $k\times\infty=\infty$ $$2^{\log_2 10\times\infty}=2^{\infty}$$

Therefore $$\infty\times 10^\infty=\infty\times 2^\infty$$