Let $\varepsilon > 0$ and $n \in \mathbb{N}$. Applying result R292: Union is smallest upper bound, we have the inclusion \begin{equation} \{ |X_n - Y| > \varepsilon \} \subseteq \bigcup_{m \in \mathbb{N} : m \geq n} \{ |X_m - Y| > \varepsilon \} \end{equation} Thus, result R2090: Isotonicity of probability measure yields the inequalities \begin{equation} 0 \leq \mathbb{P}(|X_n - Y| > \varepsilon) \leq \mathbb{P} \left( \bigcup_{m \in \mathbb{N} : m \geq n} \{ |X_m - Y| > \varepsilon \} \right) \end{equation} Since $n \mapsto X_n$ converges to $Y$ almost surely, result R2386: Characterisation of almost sure convergence using limit superior and limit inferior states that the quantity on the right-hand side, as a function of $n$, is a convergent sequence which converges to $0$ as $n \to \infty$. Thus, applying R1096: Squeeze theorem for basic sequences, we conclude \begin{equation} \lim_{n \to \infty} \mathbb{P}(|X_n - Y| > \varepsilon) = 0 \end{equation} Since $\varepsilon > 0$ was arbitrary, the claim is established. $\square$