In Carleson/Gamelin Complex dynamics p. 31 it shown that an analytic function $f$ can be linearized near an attracting fixed point. Let $\phi_n(z)=\lambda^{-n}f^n(z)$. I have trouble understanding the uniform convergence of $\phi_n(z)$. Choosing $\delta>0$ such that $\rho<1$ they bound $$|\phi_{n+1}(z)-\phi_n(z)|\leq \frac{\rho^nC|z|^2}{|\lambda|}$$ for $|z|\leq\delta$. Then they say $\phi_n(z)$ converges uniformly for $|z|<\delta$.
I see that $\phi_{n+1}-\phi_n$ converges uniformly by the M-test. Does it follow that $\phi_n$ converges uniformly?
To show $\phi_n(z)$ converges uniformly I would bound $|\phi_n(z)|$. I tried to use the triangle inequality and write $|\phi_n(z)-\phi(z)|\leq|\phi_n(z)-\phi_{n+1}(z)|+|\phi_{n+1}(z)-\phi(z)|$ but that doesn't help.