The Egyptians of anti quity expressed a fraction as a sum of fractions whose numerators we...

The Egyptians of anti quity expressed a fraction as a sum of fractions whose numerators were 1. For example, 5/6 might be expressed as

 =  +

We say that a fraction p/q, where p and q are positive integers is in Egyptian form if

where n1, n2, … nk are positive integers satisfying n­1 < n2 < … <nk.

Show that any fraction p/q, where p and q are positive integers, can be written in Egyptian form. (We are not assuming that p/q< 1.)