Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a
D1159: Probability space.
Let $Y, X_0, X_1, X_2, \dots \in \mathsf{Random}(\Omega \to \mathbb{R}^K)$ each be a
D4383: Random euclidean real number on $P$ such that
| (i) |
\begin{equation}
X_n \overset{a.s.}{\longrightarrow} Y
\quad \text{ as } \quad
n \to \infty
\end{equation}
|
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. $\square$
