Convergence on $L^1$ space

Question

I come across this problem while reading hints for Exercise 3.15 (Brezis):

Let $\Omega=(0,1)$ and sequence $f_n$ defined by $f_n(x)=ne^{-nx}$. Prove that: $$\displaystyle \int_{\Omega} \varphi f_n\rightarrow0,\forall\varphi\in C_c(\Omega)$$ Where $C_c(\Omega)$ is the space of continous function on $\Omega$ with compact support

I 've already proved this through following steps:

1. Let $K\subset\Omega$ be compact support of $\varphi$ and $x_0=\inf K\ge0$. We have K is closed.
2. Suppose $x_0=0$ then $0\in K$ (since K is closed) which contradicts to $K\subset (0,1)$. So $x_0>0$
3. Show that $f_n$ is decreasing function so we have $f_n(x)\le f(x_0)=ne^{-nx_0},\forall x\in K$
4. $\displaystyle \int_{\Omega} \varphi f_n= \int_{K} \varphi f_n+\int_{\Omega\setminus K} \varphi f_n= \int_{K} \varphi f_n$
5. Finally $$\displaystyle\bigg\vert \int_{\Omega} \varphi f_n\bigg\vert\le ne^{-nx_0} \int_{K}\vert \varphi\vert\rightarrow 0$$ Does my proof have anything incorrect? May i ask for improvement for this proof or even a better way to prove this?