TITLE: proof of Existence of Accumulation Point in $\Bbb{R}^2\setminus\{(0,0)\}$.
QUESTION [0 upvotes]: Let $\ E = \Bbb{R}^2\setminus\{(0,0)\}$
In this set, I would like to show that $(0,0)$ is an accumulation point of $E$.
Below is my proof:

For every open set $U$ which contains $(0,0)$,  there exists $\epsilon \gt 0$ such that $B(x, \epsilon) \subset U $.
Since every $B((0,0), \epsilon) $ contains point of $E$ other than $(0,0)$, it's an accumulation point.

How is this proof? Is it sufficiently rigorous? I doubt that there's more explanation about why that epsilon ball contains point of $E$ other than $(0,0)$.

REPLY [1 votes]: The first line of your proof is correct.
The second line is mere assertion. You need to show that $B((0,0),\epsilon)$ contains a point other than $(0,0)$. 
In this case, you can show that a specific point, say $(\epsilon/2,0)$,  is in $B((0,0),\epsilon)$.
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