Matrix rounding with respect to small submatrices

We show that any real valued matrix A can be rounded to an integer one B such that the error in all 2× 2 (geometric) submatrices is less than 1.5, that is, we have |aij − bij | < 1 and | ∑i+1 k=i ∑j+1 `=j (ak`− bk`)| < 1.5 for all i, j.