Give an example of an integer $K \in \mathbb{N}^{\star}$ for which the event in question Q33 occurs with the functions $f_n$ defined by
$$\left\{\begin{array}{l}
f_0 = 0 \\
f_n(x) = \ln\left(1 + \sin\left(\frac{x}{n}\right)\right) \quad \forall n \in \mathbb{N}^{\star}, \forall x \in [0,1].
\end{array}\right.$$