Let $n$ be a non-zero natural integer. We consider, on the probability space $(\mathfrak{S}_n, \mathscr{P}(\mathfrak{S}_n))$ equipped with the uniform probability, the random variable $X_n$ defined by $X_n(\sigma) = \omega(\sigma)$.
Justify that there exists a real number $C > 0$ such that, for any real $\varepsilon > 0$ and any integer $n \geqslant 1$, we have
$$\mathbb{P}\left(\left|X_n - \ln(n)\right| > \varepsilon \ln(n)\right) \leqslant \frac{C}{\varepsilon^2 \ln(n)}.$$