IT-CESENATICO 2013 Problem 3

Each integer is colored with one of two colors, red or blue. It is known that, for every finite set $A$ of consecutive integers, the absolute value of the difference between the number of red and blue integers in the set $A$ is at most $1000$ . Prove that there exists a set of $2000$ consecutive integers in which there are exactly $1000$ red numbers and $1000$ numbers blue.