Let $S$ be a subset of $[0, 1]$ consisting of a finite number of intervals. How to prove that if the total length of intervals from $S$ is greater than $0.6$ then $S$ contains two numbers such that their difference is exactly $0.1$?
If there are no two numbers in $S$ with difference exactly $0.1$, then the length of the intervals in $S$ has to be $\leq0.1$ (it can only equal $0.1$ if the interval is open) and also the difference between the upper bound of an interval with the lower bound of the consecutive interval to the right should be $\geq 0.1$. But since length of $S$ is $0.6$, there cannot be more than 5 intervals in $S$ with 4 gaps in between them, where the sum of the lengths of the gaps is $0.4$. But then length of $S$ cannot be more than $0.5$. Hence contradiction.