I worked this out myself and I was wondering if somebody could help me and see if I have made any mistakes. It's a bit of a tautology since an open disc is called an open disc but I wanted to prove this from my definition

Definition: is open

1. E is empty or

2. such that

where (open disc).

So let's prove that D(a,r) itself is open!

We want to prove the statement

such that

Proof:

Let . Then .

Choose since .

Let . Then .

Hence and we conclude that .