Let $n \geq 1$ and $P[x]$ in $E_n$ the space of monic polynomials of degree $n $. Show that $ \inf_{P \in E_n} \int_0^1 | P ( t ) | d t> 0 $.

Let $n \geq 1$ and $P[x]$ in $E_n$ the space of monic polynomials of degree $n $. Show that $ \inf_{P \in E_n} \int_0^1 | P ( t ) | d t> 0 $.

This exercise comes from excercise $30$ of this link and there is an answer using the topology of $R[X]$. I was wondering if a solution existed using more standard analysis. My idea was to look at the intervals close to $0$ and $1$ to get some bounds on the coefficients, but I couldn't really solve it explicitly. Any help would be appreciated.