Let $K\subset\mathbb{R}^n$ be a closed convex set. Let $p\in (1,\infty)$, $\delta\geq 0$ and $F_\delta:\mathbb{R}^n\rightarrow\mathbb{R}$ defined by $F_\delta(x)=(\delta^2+|x|^2)^{\frac{p}{2}}$
Note that $F_\delta$ is strictly convex, hence, we can find for all fixed $\delta$, a unique point $x_\delta\in K$ such that $\inf_{x\in K}F_\delta(x)=F(x_\delta)$
Denote $x_0=x$ and $F_0=F$. How can one show that $x_\delta\rightarrow x$.