2. (5 points) Consider the IP: max $\vec{c} \cdot \vec{x}$, such that $A\vec{x} \le \vec{b}$, $\vec{x} \ge 0$ and $\vec{x} \in \mathbb{Z}^n$, where A, $\vec{b}$ and $\vec{c}$ are all composed of positive integers. Assume that $\vec{x}^*$ is an optimal solution for the linear relaxation problem LP. Show that $[\vec{x}^*]$ is IP feasible, and its \"cost\" can be no further from optimal than $\sum_{i=1}^n c_i$. (For a real number x, the symbol $[x]$ means the largest integer less than x; if $\vec{x}$ is a vector, $[\vec{x}]$ is the vector obtained by taking [] on each component.)
