Question about Royden's proof that Lebesgue measure is countably additive.

Question about Royden's proof that Lebesgue measure is countably additive.

In Royden's book, he gives the following book for the proof that Lebesgue
measure is countable additive. I will just give a sketch of the proof.
Let $\{E_k \}_{k=1}^{\infty}$ be a collection of disjoint measurable sets.
We only need to show that $$ m(\cup_{k=1}^{\infty} E_k) \geq
\sum_{k=1}^{\infty} m(E_k) $$ since it follows by countable subadditivtiy
that $$ m(\cup_{k=1}^{\infty} E_k) \leq \sum_{k=1}^{\infty} m(E_k) $$ For
any finite subcollection of $\{E_k \}_{k=1}^{\infty}$, it follows by
monotonicty that $$ m(\cup_{k=1}^{\infty} E_k) \geq \sum_{k=1}^{n} m(E_k)
$$ for each n. What confuses me is the next statement: "Since the left
hand side of the inequality is independent of n it follows that"
$$ m(\cup_{k=1}^{\infty} E_k) \geq \sum_{k=1}^{\infty} m(E_k) $$ as
desired. Why is it true that the left hand side of this inequality is
independent of n?