This is part of Papa Rudin Chapter 2 Exercise 22.
Suppose that $X$ is a metric space, with metric $d$, and that $f:X \to [0, \infty]$ is lower semicontinuous, $f(p)< \infty$ for at least one $p \in X$. For $n=1,2,3,..., x \in X$, define
$$g_n (x)= \inf \{ f(p)+nd(x,p): p \in X \}$$
and prove that $|g_n (x) - g_n (y)| \leq nd(x,y)$.
I refer to one solution from online. But I don't think it's correct.
