I was reading a section in a Linear Algebra textbook and I have a small doubt.  I could prove that intersection of any finite collection of subspaces is a subspace. I did this by first doing it for two subspaces, and then extend it easily. But, I am not sure how it would extend to intersection of infinite collection.  My argument for a finite case involves first proving that intersection of two subspaces $W1$ and $W2$ is a subspace. Then let $W12=W1 \cap W2$, and then the same argument will work for $W12$ and $W3$, and so on.  But is the above method enough to conclude that even an intersection of an infinite collection would be a subspace.  Thanks in advance for your help.
Let $V$ be a finite dimensional vectorspace over a field $\mathbb{ F}$. It's easy to show that if $U$ and $V$ are subspaces of $V$ then $U \cap V$ is a subspace. But what if there are an infinite number of them? I have seen sometimes in pure maths that you take an arbitrary finite collection of sets and show the statement you are trying to prove is true for them to deduce it is true for the whole infinite collection. Why is this technique allowed? Can I do it for this case? Why?
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