Suppose $f(z)=\sum a_n(z-z_0)^n$ and when $|z-z_0|\leq r<R$, there is a constant C so that $|f(z)-f(z_0)|\leq C|z-z_0|$. Suppose $|z-z_0|\leq r<R$, I try to show $\left|f(z)-\sum_{m=0}^{k}a_m(z-z_0)^m\right|\leq D_k|z-z_0|^{k+1}$ with $D_k$ constant
Here is my approch: \begin{align*} |f(z)(z-z_0)^k-f(z_0)(z-z_0)^k|=|f(z)-f(z_0)||z-z_0|^k\leq C|z-z_0|^{k+1} \end{align*} And \begin{align*} LHS=\left|\sum_{n=k}^{\infty}a_n(z-z_0)^n-a_0(z-z_0)^k\right|&=\left|\sum_{n=0}^{\infty}a_n(z-z_0)^n-\sum_{m=0}^{k-1}a_m(z-z_0)^m-a_0(z-z_0)^k\right|\\ &=\left|f(z)-\sum_{m=0}^{k-1}a_m(z-z_0)^m-a_0(z-z_0)^k\right|\\ &=\left|f(z)-\sum_{m=0}^{k}a_m(z-z_0)^m+a_k(z-z_0)^k-a_0(z-z_0)^k\right|\\ &\geq\left|f(z)-\sum_{m=0}^{k}a_m(z-z_0)^m\right|-\left|(a_k-a_0)(z-z_0)^k\right|\\ \end{align*} So $\left|f(z)-\sum_{m=0}^{k}a_m(z-z_0)^m\right|\leq (C|z-z_0|+|a_k-a_0|)|z-z_0|^k\leq (Cr+|a_k-a_0|)|z-z_0|^k$
I didn't see errors on my proof but fail to get $|z-z_0|^{k+1}$ on RHS. Can anyone help, thanks in advance