Union & Intersection of Collections of Open Sets

Proof

1. Let $U_{\lambda }$ be open in $\mathbb{R}$, for $\lambda \in \mathrm{\Lambda }$. Let $x\in \bigcup _{\lambda \in \mathrm{\Lambda }}{U}_{\lambda }$. So $\mathrm{\exists }{U}_{{\lambda }_0}$ where $x\in U_{{\lambda }_0}$ which itself is open. So then $\mathrm{\exists }\epsilon >0$ such that $V_{\epsilon }(x)\subseteq U_{{\lambda }_0}\subseteq \bigcup _{\lambda \in \mathrm{\Lambda }}{U}_{\lambda }$, completing the proof.
2. Let $U_1,\dots ,U_n$ all be open sets in $\mathbb{R}$. Let $x\in \bigcap _{i=1}^n{U}_i$. Then $x\in U_i$ for each $i=1,...,n$. Each $U_i$ is open so then for each $\mathrm{\exists }{\epsilon }_i>0$ where $V_{{\epsilon }_i}(x)\subseteq U_i$. Then choose $\epsilon =min({\epsilon }_1,\dots ,{\epsilon }_n)$, which we can do since we have a finite number of them. Then clearly $\epsilon >0$ and also since it's the smallest then:

for all $i$ here, so then $V_{\epsilon }(x)\subseteq \bigcap _{i=1}^n{U}_i$ as desired.
☐