Convergence in Probability

In the setting of Probability Spaces: Notation, let $(S,d)$ be a separable metric space, let $(X_{n})_{n\in\mathbb{N}}$ be a sequence of random elements of $S$, and let $X$ be a random element of $S$. Each $d(X_{n},X)$ is a random variable (The Distance of Two Random Elements of a Separable Metric Space Is a Random Variable (i)).

(i) Convergence in probability. $(X_{n})$ converges to $X$ in probability if for every real $\varepsilon>0$

$$P\bigl(d(X_{n},X)>\varepsilon\bigr)\to0\qquad(n\to\infty),$$

convergence of real numbers as in Convergent Sequences of Real Numbers (i).