Let $D \subset \mathbb{R}^{n}$ be a convex closed set and let $f:\mathbb{R}^{n}\rightarrow \mathbb{R}$ be a convex function. Prove that if $x^{*}$ is a solution of
$$\min_{x \in D} f(x)$$
then there exists a $\beta > 0$ that makes $x^{*}$ a solution of
$$\min_{x \in \mathbb{R}^{n}} \left( f(x) + \beta \, \mbox{dist}(x,D) \right)$$
without restrictions (where $\mbox{dist}(x,D)$ is the distance between a point $x$ and the set $D$).