Bounded Lower Semicontinuous Functions are Increasing Limits of Lipschitz Functions

Let $(X,d)$ be a metric space with $X$ nonempty, let $M$ be a real number with $0\le M$, and let $f:X\to\mathbb{R}$ be lower semicontinuous on $X$ with $0\le f(x)\le M$ for every $x\in X$. For $k\in\mathbb{N}$ let $\lambda_k=\iota(k)$, where $\iota$ is the canonical map from $\mathbb{N}$ to $\mathbb{R}$, and define $f_k:X\to\mathbb{R}$ by

$$f_k(x)=\inf\bigl\{f(y)+\lambda_k\,d(x,y)\;:\;y\in X\bigr\}\qquad(x\in X).$$

1. For every $k\in\mathbb{N}$ and every $x\in X$ the infimum above exists in $\mathbb{R}$, and $0\le f_k(x)\le f(x)\le M$.

2. For every $k\in\mathbb{N}$ the function $f_k$ is Lipschitz with constant $\lambda_k$, as a map from $(X,d)$ to $\mathbb{R}$ equipped with the absolute-value metric, and is therefore continuous on $X$.

3. For every $k\in\mathbb{N}$ and every $x\in X$, $f_k(x)\le f_{k+1}(x)$.

4. For every $x\in X$ the sequence $(f_k(x))_{k\in\mathbb{N}}$ converges to $f(x)$, and $f(x)$ is the least upper bound of the set $\{f_k(x):k\in\mathbb{N}\}$.