Prove that the following is an open set.

Question

Just wondering if I am anywhere near the correct answer.

We have the set {z $\in \mathbb{C} : |z - 1| < |z + i|$}

So we let $0 < \delta < |z + i| - |z - 1|$ then use the triangle inequality to prove $\delta + |z - 1| < |z + i|$.

Thanks!

Answer 1

You are as far as $\delta$ to $0$.

Consider its geometric meaning instead: this set depicts the points closer to $1$ than to $-i$. Can you draw it?

Answer 2

So far so good.

Now you consider the open ball around your $z$ with radius $\delta /2$ and show that it is included in the region $|z-1|< |z+i|$ using the triangle inequality.