Applying (i) and (ii) repeatedly yields \begin{align*} |f(0)| &= |f^3(0) - f(0)| \leq |f^2(0)|\\ &= |f^2(0) - f^3(0)| \leq |f(0) - f^2(0)| \leq |f(0)|. \end{align*}Hence $|f(0)| = |f^2(0)|$. Now suppose $f(0) = \lambda\neq 0$ by way of contradiction; then either $f^2(0) = \lambda$ or $f^2(0) = -\lambda$. In the former case, $f^3(0)$ also equals $\lambda$, which is a contradiction. In the latter case, we have \[ 2|\lambda| = |f^2(0) - f(0)| \leq |f(0)| = |\lambda|, \]which is also a contradiction. Hence our assumption was false and $f(0) = 0$.