I am currently working through a paper on Gradient Flows in Metric Spaces by Philippe Clément and need to verify the following.
Let $(X,d)$ be a complete, metric space and $\phi : X \to (-\infty, \infty]$ be proper, lower semi continuous and bounded below. Show that $\varphi_n(y) := \inf_{z\in X} \{\phi(z) + nd(y,z)\}$ converges to $\phi$ and $\varphi_n \in \text{Lip}(X;\mathbb{R})$.
What I have tried: I have tried to show the existing of a minimizer of the $\inf$ in $\varphi_n$. Given the fact that everything there is lower semi continuous and bounded from below, I only needed convergence of a minimizing sequence which I have failed to show thus far.
What bothers me even more is that the convergence (of $\varphi_n$) should be obvious since the second term acts as some kind of penalty term if the minimizer were not to be $z = y$, but I am struggling to put that on paper.
I'd appreciate any help!
Consider a sequence of $\varepsilon$-minimizers $z_n$ of the inf in $\phi_n$, i.e., $z_n$ is such that $\phi(z_n)+nd(y,z_n) \leq \phi(y)+\varepsilon$. Because $\phi(z_n)$ is bounded from below and $\phi(z_n)\leq \phi(y)-\varepsilon$ this means that $n d(y,z)$ is bounded. In particular, $z_n\to y$. By lower semi-continuity it follows that $\phi(y)\leq \liminf \phi(z_n)-\varepsilon \leq \phi_n(y)-\varepsilon$. Because $\varepsilon$ was arbitrary, the convergence follows.
For the Lipschitz constant, let $v,w\in X$. By triangle inequality of the norm we have $$ \phi_n(v) = \inf_z \{ \phi_(z)+nd(z,v) \} \leq \inf_z \{ \phi_(z)+nd(z,w) \} + nd(v,w) = \phi_n(w)+nd(v,w) $$
Switching the roles of $v,w$ yields the result with Lipschitz constant $n$.