14th Putnam 1954

Problem B3

Let S be a finite collection of closed intervals on the real line such that any two have a point in common. Prove that the intersection of all the intervals is non-empty.




Let [A, B] be the interval with the largest left-hand endpoint, and let [a, b] be the interval with the smallest right-hand endpoint. Then since [A, B] and [a, b] overlap, we must have A ≤ b, so [A, b] is non-empty.

Now given any interval [x, y] in S, we have x ≤ A and y ≥ b, so [A, b] ⊆ [x, y].



14th Putnam 1954

© John Scholes
24 Nov 1999