Real analytic approximations which almost preserve Lipschitz constants of functions defined on the Hilbert space

Let $X$ be a separable real Hilbert space. We show that for every Lipschitz function $f:X\to \mathbb{R}$, and for every $ϵ>0$, there exists a Lipschitz, real analytic function $g:X\to \mathbb{R}$ such that $\mathrm{\mid }f(x)-g(x)\mathrm{\mid }\le ϵ$ and $\text{Lip}(g)\le \text{Lip}(f)+ϵ$.