I need some helps to solve this proof, any guidance will be greatly appreciated.
Suppose $\mathbf{x}_k$=$Prox_{\eta}\,g(\mathbf{y}_k-\eta\nabla f(\mathbf{y}_k))$, prove that: \begin{align*} g(\mathbf{x}_k) \leq g(\mathbf{y}_k) - \langle \nabla f(\mathbf{y}_k),\mathbf{x}_k-\mathbf{y}_k \rangle-\frac{1}{2\eta}\|\mathbf{x}_k-\mathbf{y}_k\|_2^2 \end{align*}
Here is my attempt: I try to first solve the proximal mapping by taking the optimal condition w.r.t. to $w\in dom(g)$, for which I got $Prox_{\eta}\,g(\mathbf{y}_k-\eta\nabla f(\mathbf{y}_k))=w=\mathbf{y}_k-\eta(\nabla g(w)+\nabla f(\mathbf{y}_k))$.
Then I try to substitute this into the definition of 1-strong convexity for function $g$: \begin{align*} &g(\mathbf{y}_k)\geq g(\mathbf{x}_k)+\langle\nabla g(x),\mathbf{y}_k-\mathbf{x}_k \rangle+\frac{1}{2}\|\mathbf{y}_k-\mathbf{x}_k\|_2^2 \\ &g(\mathbf{x}_k) \leq g(\mathbf{y}_k)-\langle\nabla g(x),\eta(\nabla g(w)+\nabla f(\mathbf{y}_k) \rangle+\frac{1}{2}\|\mathbf{y}_k-\mathbf{x}_k\|_2^2 \end{align*} But this looks like a bad approach to me and I don't how to proceed beyond that.