formula

$|(A-A_k)x|^2&=&\lim_{N\to\infty}\sum_{n=k+1}^{N}|\langle Ax,e_n\rangle|^2\\ &=&\lim_{N\to\infty}\sum_{n=k+1}^{N}|\langle x,A^*e_n\rangle|^2\quad (A^*: {\rm adjoint~operator})\\ &\leq &\lim_{N\to\infty}|x|^2\sum_{n=k+1}^{N}|A^*e_n|^2\\ &=&|x|^2\sum_{n=k+1}^{\infty}|A^*e_n|^2$