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

is open.

My attempt at a proof so far gets here:

Let

. Then

if,

So my aim now is to show that

, equivalently, that

I figure I start with

, 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.