Lipschitz continuity for a martingale (Durrett probability 4th edition 5.5.3)

Question

I want to show that for $f$ Lipschitz continuous, $X_n=\frac{(f((k+1)2^{-n})-f(k2^{-n}))}{2^{-n}}$ defines a martingale that converges almost surely in L^1. $f:[0,1]\to\mathbb{R}$. I have the following: $$E(X_{n+1}|\mathcal{F}_n)=E\left(\frac{(f((k+1)2^{-(n+1)})-f(k2^{-(n+1)}))}{2^{-(n+1)}}|\mathcal{F}_n\right)\leq E\left(K\frac{(k+1)2^{-(n+1)}-k2^{-(n+1)}}{2^{-(n+1)}}|\mathcal{F}_n\right)\leq E(K|\mathcal{F}_n)$$ where $K$ is the Lipschitz constant. My issue is with finding what $K$ is, as well as a lower bound, since what I have will at best show that this is a supermartingale. I'd appreciate any help!