  Let $X$ be a separable real Hilbert space. We show that for every Lipschitz
function $f:X\rightarrow\mathbb{R}$, and for every $\epsilon>0$, there exists a
Lipschitz, real analytic function $g:X\rightarrow\mathbb{R}$ such that
$|f(x)-g(x)|\leq \epsilon$ and $\textrm{Lip}(g)\leq \textrm{Lip}(f)+\epsilon$.
