Density of Lipschitz functions in the set of uniformly continuous functions

Question

Let $(E,d)$ be a compact metric space we consider the sup distance

$$d_\infty(f,g) = \sup_{x\in E}d(f(x),g(x))$$

In the exercise it is asked to prove that $(Lip(E), d_\infty$) set of Lipschitz functions on $(E,d)$ is dense in $(Uc(E), d_\infty$) set of uniformly continuous functions on $(E,d)$.

Moreover for given $f:E\to \Bbb R$ then construct an explicit sequence $(f_j)_j$ of Lipschitz functions converging uniformly to $f$.

In fact this exercise to be corrected next week.

So I kindly ask for a hint or trick on how I can construct such sequence.

Answer 1

Hint/trick: take $$f_j(x)=\inf_{y\in E}\{f(y)+j\cdot d(x,y)\}.$$

$$f_j\overset{Uc}{\longrightarrow} f$$