Riesz Representation Theorem from Bartle

Question

This is proof of Riesz Representation theorem from Bartle. I don't understand one step in this proof. The author shows that $\lambda$ is a measure, but I am not sure how the highlighted line shows that it is. I guess the highlighted line is related to the countable additivity, but I can't see it does. I appreciate if you give some help.

Answer 1

To prove countable additivity, it suffices to prove continuity from below, namely that if $E = \cup_n E_n$ where $E_1 \subseteq E_2 \subseteq \dots$, then $\lim_n \mu(E_n) = \mu(E)$.