Given any real number, there is a real number that is greater. a. given any real number r, there is s such that s is

In order to prove the statement, we have to find a rational number x which satisfies r  <   x   <    s.

For the number:    >    x    =    r   +  s / 2,    

We Have:

r      =     r   +  r / 2     <       r     +   s / 2    <      s   +  s /2    >    s

It remains to prove that x is rational.

Let: >       r    =   a/b        and       s     =    c/d

where: >   a, b  6   =   0, c,    and d   6   =    0 are integers.

Then:   >    x   =    r  +  s/2  

=     r        =     a/2b    +    s    =    c/2d

=           ad    +   cb / 2bd

Therefore,    where ad + cb, and 2bd 6    =   0 are integers. Therefore x is rational.

Hope that helps!!!                                     : )

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