Given a linear programming problem $\min c^Tx$ s.t. $Ax=b,\ x\geq0$. The solution to this problem is $z_0=c^T x_0$, and the solution to the dual of this problem is $\lambda^T b$. The solution to $\min c^Tx$ s.t. $Ax=b+\Delta b,\ x\geq0$ is $z_0+\Delta z=c^T (x_0+\Delta x)$, where $\Delta b$ is small enough. How to prove that $\Delta z=\lambda^T \Delta b$?
I know that $z_0=\lambda^T b$. If the solution to the dual of the new problem is $\lambda_2^T(b+\Delta b)$, we also have $\lambda_2^T(b+\Delta b)=z_0+\Delta z$. Then it suffices to prove $(\lambda_2-\lambda)^T(b+\Delta b)=0$. This is where I'm stuck. Any advice will be greatly appreciated.