Suppose that $z_0$ is a a complex number that does not lie on the real interval $[0,1]$. Next, suppose $z_n$ is a sequence of complex numbers in $C\backslash[0,1]$ that converge to $z_0$.
I am trying to show that
converges uniformly to
for all real numbers $t$ such that $0\le t\le 1$.
Really stuck. Any suggestions?