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