Second example: Irrational numbers

The set of irrational numbers is another classical example of extension by limits.

An irrational number $\zeta$ in the interval $[0,1]$ is the limit of an arbitrary sequence of digits $a_i$, that is of a sequence of rational numbers:

$$\zeta=0.a_1a_2a_3 \ldots= {a_1 \over 10 } + {a_2 \over 100 }+ {a_3 \over 1000 } + \ldots$$

In this example, the set of rational numbers is expanded by much ``larger'' set of irrational numbers.