We know that $\frac{1}{2} = 0.5$. Since $\frac{4}{9} \approx 0.44$ is less than $\frac{1}{2} = 0.5$, then 4 cannot be placed in the box. (No integer smaller than 4 can be placed in the box either.) Since $\frac{5}{9} \approx 0.56$ is greater than $\frac{1}{2} = 0.5$, then the smallest integer that can be placed in the box is 5.