Prove that the sequence of functions $\{f_{n}(z) = (1+nz)^{-1}\mid n=1,2,...\}$ converges uniformly to $f(z)=0$ for $|z| \geq r > 0$. To answer the question, for a given choice of $\epsilon > 0$, you must show how to choose $N$ such that if $n \geq N$, then $|f_{n}(z)-0|=|f_{n}(z)| < \epsilon$.
How would I begin this proof?