This seems obvious. When trying to prove that the union of algebras is an algebra there is the following difficulty. Suppose . Then it may happen that and . Even though and by themselves are closed under union, there is no reason to believe that or . But with intersection of algebras, this can't happen.
Edit: Next time, post similar questions to the Advanced Algebra subforum since it is intended for university-level questions. The current subforum is for pre-university level questions.