- Prove that there are uncountably many intervals $(a,b)$ in $\mathbb{R}, a\neq b$.
- Assume $X$ be an uncountable family of intervals. Show that there exists at least two intervals in this family that overlap.
First was not difficult. I used the arguments similar to Cantor's Diagonal Argument (used to show $\mathbb{R}$ is uncountable.)
My attempt for 2: Assume $X$ be an uncountable family of pairwise disjoint intervals, i.e. $(a_i,b_i) \cap (a_j,b_j) = \emptyset, \quad \forall i\neq j\in I$. We know there exists a rational number in each of these intervals. This implies there are uncountably many rational numbers. Contradiction, since $\mathbb{Q}$ is countable. Thus, $X$ must have at least two intervals that overlap. $\blacksquare$
Is there any problem with this reasoning?