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!!!