Exercise 1.  Consider the function   ,   where   .  

1 (c).  Use induction to show that, if all the terms of the sequence    are defined, then the sequence    is real, provided that    is real.

Solution 1 (c).

See text and/or instructor's solution manual.

If      is real then obviously      is real.  

        Assume    is real for some  ,  

then      is also real, provided  .

        Therefore, by mathematical induction,

if all the terms of the sequence    are defined, then the sequence    is real, provided that    is real.

We are done.   

Remark.  We can use Mathematica to explore this situation.

                    Some terms in the sequence    generated by     are

                    

                    Notice that the iteration produces an oscillating sequence    that is divergent.

This solution is complements of the authors.

(c) 2008 John H. Mathews, Russell W. Howell