Problem : connectedness of sets

Question

Are sets $$ A = \left\{ (x,y) \in \mathbb{R}^2 \mid -1 \leq y \leq \frac{2}{x^2 + 1} -1 \right\} $$ and $$B = \left\{ (x,y) \in \mathbb{R}^2 \mid y \geq x^2 -1 \right\} $$ connected? I think they are connected. I want to use the theorem that says that if we have a countable number of nonempty sets and their intersection is not an empty set and they are connected thet the set is connected. So I want to divide these sets into a countable number of nonempty connected sets whose intersection is nonempty but have no idea how to do it.

Answer 1

You can easily make a graph and represent the two sets. Then prove that they are pathwise-connected and thus connected.