IT-CESENATICO 1993 Problem 3

Consider an infinite chessboard whose rows and columns are indexed by positive integers. At most one coin can be put on any cell of the chessboard. Let be given two arbitrary sequences ( $a_n$ ) and ( $b_n$ ) of positive integers ( $n \in N$ ). Assuming that infinitely many coins are available, prove that they can be arranged on the chessboard so that there are $a_n$ coins in the $n$ -th row and $b_n$ coins in the $n$ -th column for all $n$ .