Before revision:  We define a sequence of x_n\in \mathbb{R}^m and invertible and symmetric m\times m matrices A_nas follows: A_n=} \nabla ^2f(x_n)+\delta (x_n) ||\nabla f(x_n)|| ^{1+\alpha = x_n-w_n, where w_n=pr_{A_n,+ A_n^{-1}  \nabla f(x_n).
After revision:  We define a sequence of x_n\in \mathbb{R}^m and invertible and symmetric m\times m matrices A_nas follows: A_n=} \nabla ^2f(x_n)+\delta _n ||\nabla f(x_n)|| ^{1+\alpha = x_n-w_n, where w_n=pr_{A_n,+ A_n^{-1} \nabla f(x_n).