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.