TITLE: Proper use of "without loss of generality"
QUESTION [7 upvotes]: I'm trying to understand exactly when I can assume something "without loss of generality in a proof." The technical explanation, I believe, is that it's allowed provided that the case that I'm omitted "reduces" to the one I prove, either through interchanging of labels or through a rather trivial extension. I can really only understand this through examples, though.
The example I have in mind, where I'm not totally sure I can use it, is as follows. Suppose I have countable (as in, finite or countably infinite) sets $X_1, \ldots, $ and want to show that $\bigcup\limits_{i \in \mathbb{N}} X_i$ is countable. I want to say "without loss of generality, suppose $X_i \neq \emptyset$ for all $i$," the justification being that if I take $I = \{j \in \mathbb{N} \mid X_j = \emptyset\}$, then I have
$$ 
\bigcup\limits_{i \in \mathbb{N}} X_i = \bigcup\limits_{i \in \mathbb{N}} X_i \setminus \bigcup\limits_{i \in I} X_i,
$$
i.e., they contribute nothing at all to the union, so having in the union is somewhat "harmless." I don't know if I'm sacrificing generality by making this assumption, though.
I would appreciate any help on understanding this.

REPLY [4 votes]: I think there is a subtle distinction between your case and one where I would use WLOG in my writings.
For me, WLOG is for when you have a number of cases that are dealt with in a nearly identical manner, so that you'd really like to treat them as a single case.  For instance, if I am dealing with a rectangular prism for some reason, I can assume WLOG that no side is longer than the length and no side is shorter than the height.  I can do that because I can reorient the box to turn six potential cases into one.
Your example is different, because we deal with empty sets in a different way than we deal with non-empty sets.  I agree with your instinct to wave them away with a single phrase as it is a trivial case.  But my personal writing style would be to say something like this:

If any of the terms of the family $X$ were empty, they would obviously contribute nothing to $\bigcup_{i\in I}X_i$.  Therefore, for the remainder of this paragraph, let us assume that every term $X_i$ is non-empty....

That being said, I doubt many people would raise their eyebrows if you used WLOG to describe this circumstance, and none would be so troubled as to let it harm their understanding of your larger argument.  But I learned that my two favorite undergraduate professors have passed on since the last time I looked, and this seems like the sort of rhetorical pedantry that they would love to point out.  ^_^