We fix $K \in \mathbb{N}^\star$ and consider a sequence of random variables $(X_n)_{n \in \mathbb{N}}$ satisfying $\mathbb{P}(X_n = -1) = \mathbb{P}(X_n = 1) = \frac{1}{2}$ for all $n \in \mathbb{N}$, distinct real numbers $x_1 < \cdots < x_K$ in $[0,1]$, and a sequence of functions $(f_n)$ of class $\mathcal{C}^K$ on $[0,1]$ with real values satisfying: (H1) the function series $\sum f_n^{(K)}$ converges normally on $[0,1]$; (H2') for all $\ell \in \llbracket 1, K \rrbracket$, the numerical series $\sum f_n(x_\ell)^2$ is convergent. Show that one of the two hypotheses (H2') or (H2) (studied in part II, where (H2) states that for all $\ell \in \llbracket 1, K \rrbracket$ the numerical series $\sum f_n(x_\ell)$ is absolutely convergent) implies the other.
We fix $K \in \mathbb{N}^\star$ and consider a sequence of random variables $(X_n)_{n \in \mathbb{N}}$ satisfying $\mathbb{P}(X_n = -1) = \mathbb{P}(X_n = 1) = \frac{1}{2}$ for all $n \in \mathbb{N}$, distinct real numbers $x_1 < \cdots < x_K$ in $[0,1]$, and a sequence of functions $(f_n)$ of class $\mathcal{C}^K$ on $[0,1]$ with real values satisfying:
(H1) the function series $\sum f_n^{(K)}$ converges normally on $[0,1]$;
(H2') for all $\ell \in \llbracket 1, K \rrbracket$, the numerical series $\sum f_n(x_\ell)^2$ is convergent.
Show that one of the two hypotheses (H2') or (H2) (studied in part II, where (H2) states that for all $\ell \in \llbracket 1, K \rrbracket$ the numerical series $\sum f_n(x_\ell)$ is absolutely convergent) implies the other.