Convergent sequences in the cofinite topology

[GridironCPJ]

How can you identify the class of all sequences that converge in the cofinite topology and to what they converge to? I get the idea that any sequence that doesn't oscillate between two numbers can converge to something in the cofinite topology. Considering a constant sequence converges to the constant, a divergent sequence to +- infinity converges to all points, a sequence that gets infinitely closer to a number converges. Am I essentially on the right track here or can anyone give me a counterexample to my claim?

 

[micromass]

Try to differentiate between:

The sequence takes on a finite number of values

and

The sequence takes on an infinite number of values.