[[Convergence concepts in probability MOC]]
# Sure convergence

**Sure convergence** is [[pointwise convergence]] on [[Real random variable|real random variables]]. #m/def/prob 
Let $X_{i} : \xi \to \mathbb{R}$ define a sequence of real random variables and $X: \xi\to\mathbb{R}$ be a random variable.
Then $X_{\bullet}$ converges to $X$ surely iff for all $\omega \in \xi$
$$
\begin{align*}
\lim_{ i \to \infty } X_{i}(\omega) = X(\omega)
\end{align*}
$$


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