# Proof of Uniform Convergence

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?

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HINT: For a fixed $n$, if you know that $|z|\ge r$, what is the largest possible value of $$\left|\frac1{1+nz}\right|\;?$$ Since convergence depends only on what happens in the long run, you may assume that $n$ is big enough so that $nr>1$.