This is an exercise from an earlier calculus 1 reading at my university:
Let $X$ be a space containing infinitely many elements. In the cofinite topology, a set $\Omega$ is open iff $\Omega = \emptyset$ or $\Omega^c$ only contains finitely many elements. Which sequences converge and what is their limit?
As I had absolutely no idea how to approach this exercise, I looked up the solution and I found it quite frustrating that I didn't even understand the solution. The writer of the solution states that for any sequence $(a_n)$ in $X$, there are three possible cases:
a) There exists no value which the sequence takes infinitely many times.
b) There exists exactly one value which the sequence takes infinitely many times.
c) There exist two or more values which the sequence takes infinitely many times.
In the first case, the writer further states that the sequence converges to any value $a \in X$, in the second case, the sequence only converges to the value taken infinitely many times and in the last case, the sequence diverges. Obviously, the writer left out a proof or even a reasoning which might help a reader to understand why this is correct.
Can anyone help me out by explaining why this holds true? Thank you very much in advance.