Sets and accumulation points

Question

I need to find 3 countably infinite sets in $\mathbb R$ such that

$(a)$ every point of R is an accumulation point

$(b)$ The set has infinitely many accumulation points, none of which are in the set

$(c)$ the set has infinitely many accumulation points all of which are in the set

Answer 1

HINT: There is a really obvious answer to (a). For (b) and (c) you could start by thinking about translates of the set $\left\{\frac1n:n\in\Bbb Z^+\right\}$.