TITLE: Can a proof be just words?
QUESTION [17 upvotes]: I suppose this is a question about mathematical convention. In a problem in Introduction to Probability by Bertsekas and Tsitsiklis, they ask the reader to prove an identity. But then their proof is mostly words:

Problem 3.* Prove the identity $$A \cup \Bigg( \bigcap_{n=1}^\infty B_n \Bigg) = \bigcap_{n=1}^\infty\big(A \cup B_n\big).$$
Solution. If $x$ belongs to the set on the left, there are two possibilities. Either $x \in A$, in which case $x$ belongs to all of
  the sets $A \cup B_n$, and therefore belongs to the set on the right.
  Alternatively, $x$ belongs to all of the sets $B_n$ in which case, it
  belongs to all of the sets $A \cup B_n$, and therefore again belongs
  to the set on the right.
Conversely, if $x$ belongs to the set on the right, then it belongs to
  $A \cup B_n$ for all $n$. If $x$ belongs to $A$, then it belongs to
  the set on the left. Otherwise, $x$ must belong to every set $B_n$ and
  again belongs to the set on the left.

In mathematics, why is this allowed? Can you say that this is more correct a proof that is, "Oh, it's obvious!" or "Just keep distributing $A$ over and over ad nauseum and you get the term on the right"?
I'm not trolling. I'm genuinely curious as to how thorough one must be when using words as proof.

REPLY [2 votes]: All proofs are words.  Display and inline equations are words, phrases, and sentences written using a significantly more precise syntax and semantics than most natural languages, but they're still words.
When you speak to tell your friend, "$x = 2$", do you say words or do you somehow switch to some other mode of communication?