### 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.

**Solution**

*Easy.*

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

jscholes@kalva.demon.co.uk

24 Nov 1999