Originally Posted by

**WonderingUser** Hi, I've been stuck on this problem for a while now and I'm scared it's just me failing with the inequalities, but it might be something more fundamentally wrong with my understanding of open sets.

From what I've learnt, a set S is open if for all z in S, there exists an r>0 such that D(z;r) is a subset of S.

My problem is here: I need to prove that

$\displaystyle S:= \left\{z \in \mathbb{C} : |z-1| < |z+i|\right\}$ is open.

My attempt at a proof so far gets here:

Let $\displaystyle z \in S$. Then $\displaystyle D\left\(z;r\right\) \subseteq S$ if, $\displaystyle \forall x \in \mathbb{C} \quad s.t \quad 0<|x|<=r, \quad z+x \in S$

So my aim now is to show that $\displaystyle z+x \in S$, equivalently, that $\displaystyle |z+x-1| < |z+x+i|$

I figure I start with $\displaystyle |z+x-1| \leq |z-1|+|x| \leq |z-1|+r$, but from there I don't know where to go.

I've been on this problem a while, and I think if I can figure this out, I should be okay with problems in the future.