Understanding convergence in probability, why do we write $P(|X_n-X|>\epsilon$) instead of $P(|X_n-X|<\epsilon$)?

Question

The definition states :
a sequence ${X_n}$ of random variables converges in probability towards the random variable $X$ if for all $\epsilon \gt 0$ : $\lim_{n\to \infty}Pr(|X_n-X|\gt \epsilon)=0$\

Why can't we write $\lim_{n\to \infty}Pr(|X_n-X|\lt \epsilon)=0$ knowing that we want the probability of $X_n-X$ to be $0$ with a large enough $n$?

Answer 1

what do you mean for "they go for first one"? It depends which book you are reading. This is from Casella Berger, one of the most reputable source and, as you can see, both definitions are given

Answer 2

$$\lim\limits_{n\to \infty}\mathbb{P}[|X_n-X|>\epsilon]=0$$

is the same as writing

$$\lim\limits_{n\to \infty}\mathbb{P}[|X_n-X|\leq \epsilon]=1$$