Define the norm $\Vert \cdot \Vert_\text{Lip}:X\to [0,\infty)$ to be the least Lipschitz constant of $f$. Prove that $(X,\Vert \cdot > \Vert_\text{Lip})$ is a Banach space.
I have already shown in the standard way that there is a pointwise limit, (called $f$) that is likely Lipschitz, but I am struggling on how to impose the Lipschitz condition. I thought I could use an $\frac{\epsilon}{3}$ type argument (as used when showing uniform continuity) before letting $\epsilon \to 0$, but obviously our $\epsilon$ will depend on some $n$, and this is just a fudge.
Thanks for your help.