Consider the sequence $x_n = (7 + (-1)^n/n, -5) \in\mathbb{R^2}$.
i). Write down the set of limits in the standard topology
ii). Prove that the sequence has no limits in $\mathbb{R^2}$ with the discrete topology
iii). Prove that every point in $\mathbb{R^2}$ is a limit point if $\mathbb{R^2}$ is given with the finite complement topology
i). My answer is (7, -5)
ii). For this one I think it involves the fact that the set {(7,-5)} is open in the discrete topology but there does not exist n st $x_n \in {(7, -5)}$
iii). Not sure how to approach this one.
For (ii) probably the easiest approach is to prove that in the discrete topology, only stationary sequences converge.
(iii) is almost tautological if you know the definitions of limit point and finite complement topology. Pick any point in $\Bbb R^2$, any neighbourhood of it, and "prove" that this neighbourhood contains all but finitely many points of the sequence $x_n$.