Let $X=U \cup V$, where $U$ and $V$ are open sets in $X$. Let $f: [0,1] \longrightarrow X$ be a loop in X. Can we find a subdivision $0=a_0 < a_1 < a_2 < \cdots < a_n=1$ of $[0,1]$ such that $f([a_{i-1},a_i])$ is contained either in $U$ or in $V$?
My book suggests that it can be done by using Lebesgue number lemma. But I cannot figure out how is that possible. Please help me in this regard.
Thank you very much.