Uniform convergent and Lipschitz functions

Mar 2006
Let \(\displaystyle (f_n)\) be a uniform convergence sequence in \(\displaystyle C^0([a,b], \mathbb {R} ) \). Suppose that each \(\displaystyle f_n\) is Lipschitz with Lipschitz constant \(\displaystyle L_n\).

a) If all of \(\displaystyle L_n\) are the same, prove that \(\displaystyle \lim _{n \rightarrow \infty } f_n \) is also Lipschitz.

b) If not all of the \(\displaystyle L_n\) are the same, must the limit be Lipschitz?

My solution:

a) Let \(\displaystyle \lim _{n \rightarrow \infty } f_n = f \), for all \(\displaystyle x,y \in [a,b] \), we wish to show that there exist some constant, say \(\displaystyle L\), we will have \(\displaystyle |f(x)-f(y)|<L|x-y|\)

Since \(\displaystyle f_n\) uniformly convergence to \(\displaystyle f\), given \(\displaystyle \epsilon > 0\), there exist \(\displaystyle N \in \mathbb {N} \) such that \(\displaystyle |f_n(x)-f(x)|< \epsilon \ \ \ \ \ \forall n \geq N, \forall x \in [a,b]\)

Now, since \(\displaystyle f_n\) are Lipschitz, we then have \(\displaystyle |f_n(x)-f(x)| \leq L_n|x-y| \ \ \ \ \forall x,y \in [a,b] \)

Now, fix \(\displaystyle n \geq N\),

Consider \(\displaystyle |f(x)-f(y)|\)

\(\displaystyle = |f(x)-f_n(x)+f_n(x)-f_n(y)+f_n(y)-f(y)|\)

\(\displaystyle \leq |f(x)-f_n(x)|+|f_n(x)-f_n(y)|+|f_n(y)-f(y)|\)

\(\displaystyle \leq \epsilon + L_n|x-y|+ \epsilon \leq L_n|x-y| \)

Therefore \(\displaystyle f\) is Lipschitz.

b) Now, suppose that \(\displaystyle L_i \neq L_j \) for some \(\displaystyle i \neq j \), we can pick \(\displaystyle L = \min L_n \), then we will still have \(\displaystyle |f(x)-f(y)|<L|x-y|\)

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!!!
Jul 2009
Rouen, France
What you did for a) is correct (except the last inequality), and you can denote \(\displaystyle L_n\) simply by \(\displaystyle L\).

For b), if it was true, then for all \(\displaystyle f\), approximate it uniformly by polynomials on \(\displaystyle [a,b]\). These functions are Lipschitz. This would mean that each continuous function on \(\displaystyle [a,b]\) is Lipschitz, which is not true, for example take \(\displaystyle f(x)=\sqrt x\).