According to wolfram alpha, this inequality holds:
$$|1-z| \leq |z|+1,$$
however, I haven't been able to figure out why. The reverse inequality (or at least the way I've applied it) gives me a lower bound on $|1-z|$ instead of an upper bound. I've tried using the regular triangular inequality and adding and subtracting like $|1-z+z-1| \leq |1-z| + |z-1|$ but I'm still stuck.