I was following the proof of the Open Mapping Theorem in functional analysis in Wikipedia, and I came across a line in the proof that did not make sense.
Some notations: $U,V$ are open unit balls in $X,Y$ respectively, and $A$ is a bounded linear transformation.
I understood to the part where if $v \in V$ then, $v \in \overline{A(\delta^{-1}U)}$. However, the article claims that this implies that for every $y \in Y$ and $\epsilon > 0$, there is some $x \in X$ such that $$\|x\| < \delta^{-1}\|y\|\text{ and }\|y - Ax\| < \epsilon.$$
I am having trouble with the aforementioned implication. Any help is greatly appreciated.