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

7 (b).  Show by example that it is not necessarily the case that    converges uniformly to    on the set  T.  

Solution 7 (b).

See text and/or instructor's solution manual.

Solution.   For all  n,  let  ,  for all   ,  where  T  is the set of complex numbers.  

        Then    converges uniformly to  ,  and    converges uniformly to    (verify these statements).  

        Show    converges uniformly to  .

Let   be any integer greater than  ,  then    and  .  

Therefore, for  ,  we have   

                        for all   .

        Show    converges uniformly to  .

Let   be any integer greater than  ,  then    and  .  

Therefore, for  ,  we have   

                        for all   .

        However, even though    converges pointwise to    (explain why),  

the convergence is not uniform (verify).  

Convergence is pointwise.

Let  .  

Fix the value  .  Now choose    to be the smallest integer satisfying  .

Then for any    we will have    and    which implies    and    which implies

                        and    .

Hence, for any    we obtain

                    

Therefore,    converges pointwise to    on the set  T.  

Convergence is not uniform.

Let  .  

Then for every positive integer  n,  if  ,  then  

                    .

Thus Statement  (7-3) is satisfied:

(7-3)            There exists an  ,  such that for all positive integers  N,
            there is some    and some  
            such that .  

Therefore,    does not converge uniformly to    on the set  T.  

We are done.   

Can you come up with a different example ?

This solution is complements of the authors.

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