Given the system of linear equations $$Ax =b,$$ where $A \in \mathbb{R}^m\times \mathbb{R}^n$ and $b \in R^m$. Let $\widehat{x}$ be the approximate solution solved by Conjugate Gradient Method and setting $\widehat{b} = A\widehat{x}$. Prove that $$\dfrac{\Vert \widehat{x}-x\Vert_2}{\Vert x \Vert_2} \le \kappa(A)\dfrac{\Vert \widehat{b}-b\Vert_2}{\Vert b\Vert_2},$$ where $\kappa(A) = \Vert A^{-1}\Vert_2 \Vert A\Vert_2$ ($\Vert A \Vert_2$ is spectral norm of matrix $A$).
My attempt: Since $Ax = b$ and $A\hat{x} = \hat{b}$ then $$\Vert \hat{x} - x\Vert_2 \le \Vert A^{-1}(\hat{b}-b)\Vert_2 \le \Vert A^{-1}\Vert_2 \Vert \hat{b}-b\Vert.$$ I intend to apply similarly like above but it makes the inequality turn to $\ge$. In addition, I also realize that I still use any step in CG algorithm. But I do not know where I have to start.