Suppose $f$ is analytic in $D_r(0)$ for some $r>1$. I want to prove that $f(z)\neq \frac{z}{z+1}$ in $D_1(0)$. This is how I tried to prove this.
Assume $f(z)= \frac{z}{z+1}$ in $D_1(0)$. Now define $a_n=-1+\frac{1}{n},n\in \mathbb{N}$. Then $a_n \in D_1(0)$.
Thus $f(a_n)=\frac{a_n}{a_n+1}$. So $\lim_{n\rightarrow\infty}f(a_n)$ should exist as $f$ should be continuous at $-1$. But, $$\lim_{n\rightarrow \infty}f(a_n) \rightarrow \infty$$ This is a contradiction so the assumption is false. For some reason I feel like this is flawed. Can someone suggest a better answer or improve this? Thanks for the help in advance