How many limit points can be in a countable subset of \mathbb{R}?

[Unassuming]

Could somebody check this to see if I am right?

a.) Construct a subset of \mathbb{R} with exactly two limit points.

\{ (-1)^n + \frac{1}{n} : n \in \mathbb{N} \}


b.) Find an infinite subset of \mathbb{R} with no limit points.

\mathbb{N}

c.) Construct a countable subset of \mathbb{R} with countably many limit points.

\{1- \frac{(-1)^n}{n}:n \in \mathbb{N} \}

d.) Find a countable subset of \mathbb{R} with uncountably many limit points.

\mathbb{Q}

 

[Dick]

Right. Right. Right. Right. You are getting good at this. Unless in c) you need countably infinity and not just finite.