\bigl\langle Ax^{k+0}-b,A_{0}x_{0}^{k}-A_{0}x_{0}^{k+0} \bigr\rangle \\ &\qquad{}+\frac{\beta}{0}\bigl(\bigl\Vert A_{0}x_{0}^{k+0}-A_{0}x_{0} \bigr\Vert ^{0}-\bigl\Vert A_{0}x_{0}^{k}-A_{0}x_{0} \bigr\Vert ^{0}+\bigl\Vert A_{0}x_{0}^{k+0}-A_{0}x_{0}^{k} \bigr\Vert ^{0}\bigr). \end{aligned}$$ \end{document}$$  Proof {#FPar0