4. For $n \in \mathbb{N}$, a function of $\mathscr{P}_n$ is said to be monic when the coefficient of its term of degree $n$ is 1. We denote $\mathscr{P}_n^u$ the set of these functions. We denote $C_n = \operatorname{Inf}\left\{\int_0^1 |f(t)|\, dt \mid f \in \mathscr{P}_n^u\right\}$. a. Show that $C_n > 0$. b. For $n \geq 2$, prove that if $x \in O_{n-1}$ $$(x_{n-1})^{n+1} \leq \frac{1}{C_n}\left[Y_1(x) + \sum_{i=1}^{n-2} (-1)^i (Y_{i+1}(x) - Y_i(x))\right]$$ c. Verify that the map $Y$ extends continuously to the closure of $O_{n-1}$. d. Show that if $K$ is a compact subset of $\mathbb{R}^{n-1}$ contained in $U_{n-1}$, $Y^{-1}(K)$ is compact.
4. For $n \in \mathbb{N}$, a function of $\mathscr{P}_n$ is said to be monic when the coefficient of its term of degree $n$ is 1. We denote $\mathscr{P}_n^u$ the set of these functions. We denote $C_n = \operatorname{Inf}\left\{\int_0^1 |f(t)|\, dt \mid f \in \mathscr{P}_n^u\right\}$.\\
a. Show that $C_n > 0$.\\
b. For $n \geq 2$, prove that if $x \in O_{n-1}$
$$(x_{n-1})^{n+1} \leq \frac{1}{C_n}\left[Y_1(x) + \sum_{i=1}^{n-2} (-1)^i (Y_{i+1}(x) - Y_i(x))\right]$$
c. Verify that the map $Y$ extends continuously to the closure of $O_{n-1}$.\\
d. Show that if $K$ is a compact subset of $\mathbb{R}^{n-1}$ contained in $U_{n-1}$, $Y^{-1}(K)$ is compact.