Exercise 7.  Suppose that the sequences of functions    and    converge uniformly on the set  T.  

7 (a).  Show that the sequence    converges uniformly on the set  T.  

Solution 7 (a).

See text and/or instructor's solution manual.

Solution.   Let us say that    converge uniformly on  T  to    respectively.  

Let    be given.   

The uniform convergence of    means there exists an integer    such that    implies  

                        for all   .  

Likewise, there exists an integer    such that    implies  

                        for all   .  

If we set  ,  then for  ,

                         

for all   .  

Therefore,    converges uniformly to    on the set  T.  

This solution is complements of the authors.

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