Let $X\subset\mathbb{R}^d$ be an unbounded closed set and $C_0$ is the space of all continuous functions $h: X\to \mathbb{R}$ with compact support. I'm searching for the sequence $\{ h_n \} \subset C_0$ such that $$0 \leq h_1 \leq h_2 \leq h_3 \ldots, \lim_{n \to \infty} h_n(x) =1 \quad \text{for} \; x \in X.$$
Is there any example of such sequence? Maybe for $X=\mathbb{R}$? I'm reading Lasota and Mackey's book "Chaos, Fractals, and Noise" and they didn't include any explanation.
Because $X$ is closed, and the intersection between closed and compact is compact, we can look for functions $H_k:\mathbb R^d\to\mathbb R$. For instance we can take $h_k$ to be one on the ball of radius $k$ centered at the origin, and with support inside the ball of radius $k+1$. For instance, $$ h_k(x)=\begin{cases} 1,&\ |x|\leq k\\[0.3cm] k+1-{|x|},&\ k<|x|\leq k+1\\[0.3cm] 0,&\ |x|>k+1\end{cases} $$