# Convergence of $(x_n)_{n\in N} = (\frac{1}{n})_{n\in N}$ in different topologies [closed]

Consider $$\mathbb{R}$$ equipped with a range of different topologies: discrete; indistcrete, Euclidean, cofinite, cocountable and lower limit. Suppose we have a sequence of real numbers given by $$(x_n)_{n\in N} = (\frac{1}{n})_{n\in N}$$. I'm trying to solve these questions on convergence in these different topologies: Does this sequence converge? If yes, to what points in $$\mathbb{R}$$?

I've found the following solutions:

• Discrete: does not converge;
• Indiscrete: convergence to all $$x \in \mathbb{R}$$;
• Euclidean: converges to $$0$$;
• Cofinite: convergence to all $$x \in \mathbb{R}$$;
• Cocountable: does not converge;
• Lower limit: converges to $$0$$.

Are these results correct? I am not looking for a full proof or explanation in each case, as I think I am capable writing it out. Just wondering if my rough intuition is correct :)

• All the answers seem correct to me. :) Jun 5 at 12:34
• @SangchulLee great! thanks! Jun 5 at 12:39
• I’m voting to close this question because it was answered in the comments. Jun 12 at 9:04