The question i'm asking arises from lemma $12$ of Laver's proof for the consistency of Borel's conjecture. It has some set theoretic assumptions but i doubt they are needed in the proof. And since i'm not really good with analysis i would be really glad if someone could point me in the right direction.
Let $I_n$, $n \lt \omega$, be closed intervals in $[0, 1]$ such that for all $n$, there exists some $0 \le m \le n-1$ such that $I_n = [m/n, (m+1)/n]$. Then there exists an infinite descending subsequence of the above intervals.
My semi-idea: Let $I_n = [a_n , b_n]$. Then the $a_n$'s have either an increasing infinite subsequence or a decreasing infinite subsequence. WLOG assume the $a_n$'s themselves are increasing. Since $[0, 1]$ is bounded, the $a_n$'s converge to a real $u$. By the same argument we may assume the $b_n$'s are decreasing and converge to the same real $u$ because the length of $I_n$'s is decreasing. Now here i wanted to somehow consider an $\epsilon \gt 0$ and find intervals included in eachother but i'm failing to go further. Any help would be appreciated.