I want proof of the following statement
"Intersection of collection of algebras is also an algebra"...
Please help
Do you mean algebras over a field or algebraic structures? Are you sure these topics belong in an undergraduate subforum?
This seems obvious. When trying to prove that the union of algebras is an algebra there is the following difficulty. Suppose $\displaystyle A,B\in\mathcal{F}_1\cup\mathcal{F}_2$. Then it may happen that $\displaystyle A\in\mathcal{F}_1$ and $\displaystyle B\in\mathcal{F}_2$. Even though $\displaystyle \mathcal{F}_1$ and $\displaystyle \mathcal{F}_2$ by themselves are closed under union, there is no reason to believe that $\displaystyle A\cup B\in\mathcal{F}_1$ or $\displaystyle A\cup B\in\mathcal{F}_2$. 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.