A set is countable if either it is finite or it has the same size as $N$
$\text{Proof By Contradiction}$ :
If a function existed between $N$ and $R$ . For correspondence, f must pair with all the members of $N$ with all the members of $R$. We somehow find some $x$ which is not paired with any real number.
Suppose the correspondence f exist.
$f(1)=10.121...,f(2)=5.876...$ and so on.
| $n$ |
$f(n)$ |
| $1$ |
$10.\color{orange}{1}21...$ |
| $2$ |
$5.8\color{orange}76...$ |
|
...
|
... |
But some numbers such as $0.199...$ and $0.200...$ are equal even there decimal representations are different
We avoid this problem by not selecting the digits $0$ and $9$ during construction of $x'$. For $x'$ choose those digits which are different like for $1\rightarrow2,7\rightarrow4,...$
$x=0.17... \space \space x' =0.24... $ so by above table we know that x is not $f(n)$ for any $n$.
so "Set of all real numbers are not countable".
