I am beginning to learn measure theory, and I have a basic doubt regarding to measure of union of sets, and limits.
Let $\mathcal{A}$ be a $\sigma$-algebra on $X$, and $\mu$ be a finite measure on $\mathcal{A}$. Suppose that $A_1, A_1, A_3, ... $ are elements of $\mathcal{A}$, and further we know $$ \mu\left(\bigcup_{i=1}^{n}A_i\right)=\sum_{i=1}^{n} \mu(A_i) $$ for every positive integer $n$. How can we conclude the following? $$ \mu\left(\bigcup_{i=1}^{\infty}A_i\right)=\sum_{i=1}^{\infty} \mu(A_i) $$
My idea: Let $E_k=\bigcup_{i=1}^{k} A_i$. Then, $E_1\subseteq E_2\subseteq E_3\subseteq\cdots$ Also, $\bigcup_{k=1}^{\infty}E_k=\bigcup_{i=1}^{\infty} A_i$. Then, using continuity from below argument (Theorem 1.8 (c) in Folland), we get $$ \sum_{i=1}^{\infty}\mu(A_i)=\lim_{n\to\infty}\mu\left(\bigcup_{i=1}^{n} A_i\right)=\lim_{n\to\infty} \mu(E_n) = \mu\left(\bigcup_{i=1}^{\infty}E_n\right)=\mu\left(\bigcup_{i=1}^{\infty}A_n\right) $$
Assuming that my argument is correct (is it actually?), I would like to know whether this is the standard/canonical method for this basic problem?