Help Me Understand: Proof that Finite Intersection of Open Sets is Open

Question

The proof is here: (link).

I don't see how the third line (starting with Thus: $\exists \epsilon_i$...) is justified. That is: just because $x \in U_i$, for all $i$, how do I know that a neighborhood of $x$ is in $U_i$, for all $i$?

Answer 1

In the context of a metric space, a subset $U$ is open when for any $x$ in $U$, one can find an $\epsilon$-ball centered at $x$ and contained in $U$.

The third line applies this definition to find a collection of open balls centered at $x$ and contained in the respective $U_i$.

Answer 2

This is just by definition: the $U_i$ are open, so there is an $\epsilon_i$-ball around $x$ that is contained in $U_i$.