It is true that $C_{0}^{\infty}[a,b]$, (the space of all smooth functions f with the property that f and all its derivatives vanish at a and b) is dense in $L^p[a,b]$ with $1\leq p< \infty$ ?
How do I prove it? I know that $C_{0}^{\infty}(\mathbb{R})$ is dense in $L_{p}(\mathbb{R})$ with $1\leq p< \infty$, but I don't know if this can help me.