Proof that " If $\mu$ is continuous from below at every set $E \in $ a ring, then $\mu$ is $\sigma$-additive."

Question

Proof:

I don't understand two things about this proof:

1. How does the set $E = \cup_{i=1}^{\infty}E_{i}$ exist in $\mathcal{R}$? as a ring $\mathcal{R}$ is closed under finite union.

2. $\{F_{n}\} = \cup_{i=1}^{n} E_{i} \in \mathcal{R}$ is a monotone increasing sequence, as the limit of $\{F_{n}\} = \cup_{i=1}^{n} E_{i}$, how does this limit of $\{F_{n}\} = \cup_{i=1}^{\infty}E_{i} $ ?

I think I might have misunderstood something, and hope someone can help me out. Thanks in advance.