Exercise 13.  Complete the proof of Theorem 4.1.   

In other words, suppose that   ,   where      and   .  

Prove that   .  

Solution 13.

See text and/or instructor's solution manual.

Answer.   Duplicate the part of the Theorem 4.1 that shows  ,  but replace    with    and  .  

Solution.   First we will assume that   ,   is true, and from this deduce the truth of   .

Let    be an arbitrary positive real number.

To establish  ,  we must show that there is a positive integer    such that the inequality    holds whenever  .  

        Since we are assuming      to be true, we know according to Definition 4.1 that

there is a positive integer    such that      if   .

Recall that      is equivalent to the inequality   .  

Thus, whenever  ,  we have  

                    ,  

and this proves  .   

This solution is complements of the authors.

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