This is one of the problem I have been working from Velleman's
How to Prove it book:
Theorem: Suppose A, B, and C are sets and A ⊆ B ∪ C. Then either A ⊆ B or A ⊆ C. Is the theorem correct?
Proof. Let x be an arbitrary element of A. Since A ⊆ B ∪ C, it follows that either x ∈ B or x ∈ C.
Case 1. x ∈ B. Since x was an arbitrary element of A, it follows that ∀x ∈ A(x ∈ B), which means that A ⊆ B.
Case 2. x ∈ C. Similarly, since x was an arbitrary element of A, we can conclude that A ⊆ C. Thus, either A ⊆ B or A ⊆ C.
After reading the proof, I was pretty much convinced that everything is fine unless I saw the answer where they give the counterexample of this theorem. Now even seeing the counterexample, when I read this proof I'm not able to find any errors in it ? Can somebody point me out in the right direction ?