Exercise 19.  Let     and  .  Show that  .  

Solution 19.

See text and/or instructor's solution manual.

Note:  Theorem 2.2 is of no use here because you don't know whether    exists.   Give an    argument.  

Solution.  Let    be given.  

Since  ,  there is some number    such that    whenever  .   

Now use the hypothesis   with the above result that    whenever  .   

Hence,  if  ,  then  ,  so that  .

Therefore  .  

This solution is complements of the authors.

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