For each $n \in \mathbb{N}$ a closed interval $[x_n, y_n]$ is given. Assume that $[x_m, y_m]\, \cap \, [x_n, y_n] \neq \emptyset$ for all $m,n \in \mathbb{N}$. Show that $\bigcap_{n=1}^{\infty} [x_n, y_n] \neq \emptyset$.
What I've done is, assumed for contradiction that $\bigcap_{n=1}^{\infty} [x_n, y_n] = \emptyset$ which implies the existence of some disjoint intervals, i.e: that there exists $m,n \in \mathbb{N}$ such that $[x_m, y_m] \, \cap \, [x_n, y_n] = \emptyset$ which contradicts our hypothesis.
Now, I think this is wrong because it (i) makes the problem far too easy and (ii) infinities are fishy, I'm not convinced that the infinite intersection of intervals being empty implies that there exists at least two disjoint intervals.
Question: Is my "proof" completely off the right train of thought? How do I fix it?