Connectedness of $\mathbb R$

Question

Let $\mathbb R$ denote the real number space, and let sets A and B be closed and nonempty such that $\mathbb {R} \subset A \cup B $, why is it true that due to the connectedness of $ \mathbb{R} $, $A \cap B \neq \emptyset $?

Answer 1

If they don't intersect, since they fill R, they are complementary, hence open and closed. Connectedness implies A or B to be empty.

Answer 2

Your assumptiom is $\mathbb{R} \subset A \cap B$.

But $\mathbb{R} \neq \emptyset$ so $\emptyset \neq \mathbb{R} \subset A \cap B$ imply $A \cap B \neq \emptyset$. Note that we didn't use connectivity of $\mathbb{R}$.