In Evans's Partial Differential Equations, the following argument is made:
Choose a sequence $\{f_m\}$ of smooth functions in $C_c^\infty(U)$ with $U=(0,2)$ satisfying $$ 0\leq f_m\leq 1,\quad f_m(1)=1,\quad f_m(x)\to 0\quad \textrm{for all }x\neq 1. $$
This looks something similar to the approximation of identity. But we don't have $\int f_m=1$ here.
Let $f$ the following function :
$$f:x \mapsto \left\{ \begin{matrix} 1 && \text{if } x=1,\\ 0 && \text{otherwise.} \end{matrix}\right.$$
We want to find $(f_m)$ a sequence of smooth functions satisfiying the properties you want and that converges pointwise to $f$.
According to this wikipedia page, the following function is $C_c^\infty(\mathbb{R})$:
$$\psi:x \mapsto \left\{ \begin{matrix} e\exp\left( - \frac 1 {1-x^2} \right) && \text{if } |x|\leq 1,\\ 0 && \text{otherwise.} \end{matrix}\right.$$
The only point sent to $1$ is $0$ and the function is between $0$ and $1$. This can be proved by elementary calculations so I am not giving any more details. The next step is to translate it and modifying the length of its support.
$\psi(2(1-x))$ satisfies being compactly supported in $(0,2)$, being between $0$ and $1$ and sending $1$ to $1$ (and only $1$ is sent to $1$, this is important for the next step).
Then we have everything we need to construct our sequence.
Then the sequence of function defined by $f_n(x)=\left(\psi(2(1-x))\right)^n$ satisfies all the properties.
PS : Note that $\psi(n(1-x))$ works too.
