Outer Measure of a Countable Collection of Sets This comes from Royden and Fitzpatrick's Real Analysis:

Proposition 3 If $\{E_k\}_{k=1}^\infty$ is any countable collection of sets, disjoint or not, then
$m^*(\bigcup_{k=1}^\infty E_k) \le \sum_{k=1}^\infty m^*(E_k)$

Note, $m^*(A)$ denotes the outer measure of $A$. There are a few parts of the proof I am having trouble with. Here is the first

For each natural number $k$, there is a countable collection $\{I_{k,i} \}_{i=1}^\infty$ of open, bounded intervals for which $E_k \subseteq \bigcup_{i=1}^\infty I_{k,i}$.

This seems to rely on there being a open cover for any set of real numbers. Is this true because $\mathbb{R}$ can be written as $\bigcup_{n=1}^\infty (-n,n)$, a countable union of open, bounded sets?
Here is another part of the proof giving me trouble:

Now, $\{I_{k,i} \}_{k,i=1}^\infty$ is a countable collection of open, bounded intervals that cover $\bigcup_{k=1}^\infty E_k$: the collection is countable since it is a countable collection of countable collections. Thus, by definition of the outer measure,
$m^* \left( \bigcup_{k=1}^\infty E_k \right) \le \sum_{k,i=1}^\infty \ell (I_{k,i}) =...$

How did they obtain this inequality? The most I can get is
$m^* \left( \bigcup_{k=1}^\infty E_k \right) \le m^* \left( \bigcup_{k,i=1}^\infty I_{k,i} \right)$,
by the monotone property. Perhaps we need to show that $m^* \left( \bigcup_{k,i=1}^\infty I_{k,i} \right) \le \sum_{k,i=1}^\infty \ell(I_{k,i})$, which would be a special case of the theorem we are trying to prove.
EDIT: This is in response to Dunham's answer, but everyone else is at liberty to chime in:
Regarding part 1, yes it might be only one example of a collection, but it suffices to prove that every set of real numbers has an open cover, right? Regarding part 2, I know that the definition of the outer measure involves the infimum, that's precisely what is confusing me. Why wouldn't we have $\sum_{k,i=1}^\infty \ell (I_{k,i}) \le m^* \left( \bigcup_{k=1}^\infty E_k \right)$ instead?
 A: Part 1: You are essentially correct. You have shown one example of a collection of intervals covering $E_k$. 
Part 1-Edit: If you look at the proof in the book, they actually ask for an open cover satisfying an additional property: 
$\sum_{k=1}^\infty \ell(I_{k,i}) \leq m^*(E_k)+\epsilon/2^k $. 
The existence comes from the definition 
$m^*(E_k)= \inf \left\{ \sum_{k=1}^\infty \ell(\bar{I}_{k,i}) | E_k \subset \cup_{i=1}^{\infty} \bar{I}_{k,i} \right\}$.
Recall that if the infimum of a set of numbers is $a$, then something between  $a$ and $a+\epsilon$ must be in the set.
Part 2: Look back to the definition of an outer measure. It is the infimum, over all coverings by intervals, of the sum of the lengths of those intervals. $\{I_{k,i}\}$ is one example of a covering, hence the inequality. 
Part 2-Edit
Every element of a set is at least as large as the infimum. $\{I_{k,i}\}_{k,i}$ is a cover of $\cup_{k} E_k$, so $\sum_{k,i} \ell(I_{i,k})$ is an element of the set in the definition of $m^*(\cup_{k} E_k)$.  
