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. .