MathDB

On a line, there are

n

closed intervals (none of which is a single point) whose union we denote by

S

. It's known that for every real number

d

0<d\le 1

, there are two points in

S

that are a distance

d

from each other. (a) Show that the sum of the lengths of the

n

closed intervals is larger than

\frac{1}{n}

. (b) Prove that, for each positive integer

n

, the

\frac{1}{n}

in the statement of part (a) cannot be replaced with a larger number.

Problem Statement