We assume that for all $d \geqslant 0$, $\mathbb{P}(X \in d\mathbb{Z}) < 1$, and that $g$ is of class $\mathscr{C}^1$, with support in $[0, K]$ with $K > 0$. We admit that there exists a subsequence $\left(t_k\right)_{k \geqslant 0}$ of $\left(y_n\right)_{n \geqslant 0}$ such that the sequence of functions $\left(\xi_k\right)_{k \geqslant 0}$ defined by $$\xi_k : \mathbb{R} \rightarrow \mathbb{R}, \quad t \mapsto \xi_k(t) = f'\left(t + t_k\right)$$ converges uniformly on every segment of $\mathbb{R}$ to a function denoted $\xi$. Show that $\xi$ is constant, equal to $c$.
We assume that for all $d \geqslant 0$, $\mathbb{P}(X \in d\mathbb{Z}) < 1$, and that $g$ is of class $\mathscr{C}^1$, with support in $[0, K]$ with $K > 0$. We admit that there exists a subsequence $\left(t_k\right)_{k \geqslant 0}$ of $\left(y_n\right)_{n \geqslant 0}$ such that the sequence of functions $\left(\xi_k\right)_{k \geqslant 0}$ defined by
$$\xi_k : \mathbb{R} \rightarrow \mathbb{R}, \quad t \mapsto \xi_k(t) = f'\left(t + t_k\right)$$
converges uniformly on every segment of $\mathbb{R}$ to a function denoted $\xi$. Show that $\xi$ is constant, equal to $c$.