Exercise 5, § 2

Let l be a number > 0, and suppose that to each point x ∈ R corresponds an open interval I(x) with centre x and length ≤ l. Show that every compact interval $[a, b]$ can be covered by a finite number of intervals $I(x_i)$, the sum of whose lengths is $\leq l + 2(b - a)$. (Prove that if the proposition is true for each interval $[a, x]$ such that $a \leq x < c$, then there exists $d > c$ such that it is true for each interval $[a, y]$ such that $a \leq y < d.$) If $l = (b - a)/n$, where $n$ is an integer $\geq 1$, show that the result cannot be improved.