# How to subtruct countable sequence from countable sequence

In a classic proof that the set of sequence points together with its limit is compact, we use the statement that any neighbourhood of a limit point contains infinite number of sequence points, hence we need only finite number of open sets to cover the rest of the sequence. That is, we state,, that countable minus countable is finite. On the other hand, if we subtract from countable set of natural numbers all odd numbers, then the set of all even numbers is countable. Thus, countable minus countable is countable. How to deal with cardinality in this case? What am I missing? Marina

• That is the definition of a limit point. if the sequence is $a_n$, given any neighborhood you can find an $N$ such that for all $n \gt N,$ the $a_n$ is within the neighborhood. That means there are at most $N$ that are outside. – Ross Millikan Dec 2 '16 at 4:54