When are Lipschitz constant uniformly bounded? I have a sequence of Lipschitz functions $f_i$ which converges uniformly to a Lipschitz function $f_0$. I also can make it the case that the domain for these functions are compact. Are these conditions enough to guarantee that the Lipschitz constants for $f_i$ are bounded?
EDIT: Based on Anthony Carapetis's quick comment, what if all functions mentioned here are convex?

Let $f_i : [0,1] \to \mathbb R$ be a piecewise linear function that is $0$ on $[0,1-1/i^2]$ and then increases from $0$ to $1/i$ on the interval $[1-1/i^2, 1]$. These functions are convex and uniformly convergent to $0$, but the Lipschitz constant of $f_i$ is $i$. 
If you want the convergence to imply a uniform Lipschitz bound, you need the convergence to be in a stronger sense - the most natural would be Lipschitz convergence, where we require not only $f_i - f \to 0$ uniformly but also that the Lipschitz constant of $f_i - f$ converges to zero.
