I was learning about the convergence in probability. I'm unsure if looking at the epsiolon-delta definitions below which captures better the convergence in probability correctly.
$$ \forall \epsilon,\delta>0 \quad \exists n>N \qquad S.T \qquad Pr(|X_n -X|>\epsilon)<\delta $$
$$\forall \epsilon>0\quad\exists n>N \qquad S.T \qquad Pr(|X_n -X|>\epsilon)=0 $$
The usual definition of convergence in probability is $$ X_n \to_{p} X \iff \forall \epsilon > 0, \lim_{n \to \infty} P(|X_n - X| > \epsilon) = 0 $$
The limit is the "basic real analysis" sequence definition, since $P(|X_n - X| > \epsilon)$ is just a sequence over $n \in \mathbb{N}$. So using that definition, we have
$$ \forall \epsilon > 0, \lim_{n \to \infty} P(|X_n - X| > \epsilon) = 0 \iff \forall \epsilon > 0, \forall \delta > 0, \exists N \in \mathbb{N} \; \text{s.t.} \; \forall n \geq N, P(|X_n - X| > \epsilon) < \delta $$
This matches basically with the first one you wrote.
