Consider the series

Where and are integers lying between 0 and 9.

From monotonically non-decreasing bounded partial sums, if follows that (*) is convergent. The series (*) is often abbreviated as (called decimal representation) and is used for the real number which is the sum of the series (*). Thus every decimal representation represents a real number.

On the other hand, every real number has a unique decimal representation. Consider any . Then , such that .

Next we have integers lying between 0 and 9 such that

,

,

…………………………………….

…………………………………….

Hence, .

Since and converges, therefore, . i.e. .

It is simple to see that the decimal representation of every rational number is either terminating or recurring. Every non-recurring infinite decimal represents an irrational number.