Let be a sequence in R

For (1), every open set containing a point in should contain all but finite elements of , since an open set in a cofinite topology is defined as X\O is finite, where X is a set with a topological space T and O is an open set in a cofinite topology.

For an arbitrary open set O containing a point should contain all memebers of for n>=N where N is a positive integer in a cofinite topology. That means, every sequence with distinct points (including elements repeated finite times) converges to every point in the real set in a cofinite topology.

For (2), {1,1/2,1/3,...,1/n}, n=1,2,.., .

For (3), {1,2,3,....,n}, n=1,2,..., .