Expanding on some of the comments, I'd say it would be inappropriate to give a reference for this fact. It is a standard result that is very easy to prove, and included in every introductory undergrad textbook and course in probability.
Generally, results that are so well known can be cited by name (if at all) without giving a specific reference, e.g. "by the fundamental theorem of calculus". In this case, what I would write would depend on what clarification was called for by context. One option would be to write "because the $X_i$ are independent", if that fact may have been forgotten at this point in the argument. If you have a lower opinion of your reader, you could say "because a sum of independent normals is normal", but if your audience is researchers, they may find it patronizing.
I occasionally see papers that make a big deal out of using a standard fact, and give a reference to a standard textbook. Rightly or wrongly, this tends to make me question the author's expertise.
You might also ask yourself: if a reader has little enough experience with probability that this fact is not familiar, will he or she have any chance of following the rest of the paper?
If your audience is undergraduate students, a reference could possibly be appropriate: pull any introductory probability text off your shelf and cite it. But again, think about how many readers would be materially helped by such a reference.