Uniform convergent and Lipschitz functions

Let be a uniform convergence sequence in . Suppose that each is Lipschitz with Lipschitz constant .

a) If all of are the same, prove that is also Lipschitz.

b) If not all of the are the same, must the limit be Lipschitz?

My solution:

a) Let , for all , we wish to show that there exist some constant, say , we will have

Since uniformly convergence to , given , there exist such that

Now, since are Lipschitz, we then have

Now, fix ,

Consider

Therefore is Lipschitz.

b) Now, suppose that for some , we can pick , then we will still have

I'm pretty sure I'm wrong somewhere here, because there is no way this problem can be this easy. Any help please? Thank you!!!

Re: Uniform convergent and Lipschitz functions

What you did for a) is correct (except the last inequality), and you can denote simply by .

For b), if it was true, then for all , approximate it uniformly by polynomials on . These functions are Lipschitz. This would mean that each continuous function on is Lipschitz, which is not true, for example take .