For all but finitely many $n \in \mathbb N$ In my book I have the following theorem:

A sequence $\langle a_n \rangle$ converges to a real number $A$ if and only if every neighborhood of $A$ contains $a_n$ for all but finitely many $n \in \mathbb N$.

Can anyone clarify what the phrase, "All but finitely many", means?

In your sentence, "but" (is not a conjunction, but a preposition and) means "except". At first, I didn't know that the word "but" has two meanings and thus I had exactly the same doubt as you.
Notice that the bold word "but" above (is a conjunction and) doesn't have the same meaning as in your sentence.
