Construct a sequence of functions $\phi_n \in C_{c}^{\infty}(\mathbb{R})$, $0 \le \phi_{n}(x) \le 1$ $\forall$ $x \in \mathbb{R}$ such that
$\phi_n(x)=1$ if $x \in (a,b) $ and $\phi_n(x)=0$ if $x<a-\frac{1}{n}$ or $x>b+\frac{1}{n}$
prove that: $\phi_n \rightarrow \chi_{(a,b)}$ in $L^{p}$
i could do if $\phi_n \in C_{c}^{k}$, but with this problem I have trouble. i tried to construct $\phi_n$ many times but all of them are fail
For each $n\in \mathbb N$ we can do the following. There exists a smooth $f$ with support in $[a-1/n,a]$ with $f>0$ on $(a-1/n,a)$ such that $\int_{a-1/n}^a f = 1.$ And there exists a a smooth $g$ with support in $[b,b+1/n]$ with $g<0$ on $(b,b+1/n)$ such that $\int_{b}^{b+1/n} g = -1.$
Define
$$\phi_n(x) = \int_{-\infty}^x (f(t) + g(t)\,dt.$$
Then $\phi_n$ is smooth, with support in $[a-1/n,b+1/n],$ such that $0<F<1$ on $(a-1/n,a),$ $F=1$ on $[a,b],$ and $0<F<1$ on $(b,b+1/n).$ The functions $\phi_n$ do what you want.