Let $ x ∈ ℝ^n $ and $ ε > 0 $. Prove that the open ball $ B_ε(x) $ is an open set (rigorously!).
I am having trouble with the problem above. If I am understanding it correctly I should be able to set $ x ∈ ℝ^n $ and $ ε > 0 $. Then the open ball $ B_ε(x) $ centered at $ x $ of radius $ ε $ is the set {$ x ∈ ℝ^n | d(x,x_o) < ε $}. If I am correct thus far then a subset $ u\subseteqℝ^n $ is open if $ \forall$ $ x \in u$, $ \exists$ $ ε>0 $ such that $ B_ε (x)\subseteq u $.
This is as far as I can get. I do not know if this is all I have to do or if I need to do something else? If someone could Help out I would really appreciate it!
Let $y\in B_\epsilon(x)$. Then $d(y,x)=r<\epsilon$. Now take any $t\in (0,\epsilon-r)$. Can you show that $B_t(y)\subseteq B_\epsilon(x)$?