sums of rows and columns in a m times n matrix

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A nonnegative number is written in each of the squares of an $m \times n$ grid with $n>m$. (m being the number of rows and n being the number of columns) To prove: If each column contains a positive number, then there is always a square in the grid, for which the sum of the numbers in the row of the square is greater then the sum of the numbers in the column of the square.
I had a really hard time thinking about this problem over the last few weeks and I have no ideas. Any suggestions?

[miarose7878]

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