measure theory - Existence of non decreasing sequence of continuous functions aproximating $f$ in $L_p(0,infty)$

I know that the continuous functions $f:(0,\infty) \rightarrow R$ are dense in $L_p(0,\infty)$, with respect to the norm $|| \space||_p$. Therefore, if $f\in L_p(0,\infty)$ then there exists a sequence of continous functions $\{f_n\}$ in $L_p$ such that $f_n \rightarrow f$. I'm wondering if there exists a sequence that does this, but also is non-decreasing, meaning $f_{n+1}\geq f_n$ pointwise por each $n$. I believe this to be true, but I haven't been able to prove it. So, is this true? If it is, I would appreciete any tips on how to prove it.

Thanks!