Equivalent definitions of $X_n = O_p(1)$

Question

I have seen the following definitions for $X_n = O_p(1)$:

1. $\forall \epsilon>0$, $\exists \delta_{\epsilon}$, $\exists N_{\epsilon}$ such that $P(|X_n| > \delta_{\epsilon}) \le \epsilon$ for all $n > N_{\epsilon}$.
2. $\forall \epsilon >0$, $\exists M_{\epsilon}$ such that $\lim \sup_{n \rightarrow \infty} P(|X_n| > M_{\epsilon}) \le \epsilon$.

I have two questions:

Why are these two definitions equivalent to each other?

Is it also true that another equivalent definition for $X_n=O_p(1)$ is $\exists M$ such that $\lim_{n \rightarrow \infty} P(|X_n| \ge M) = 0$?

Answer 1

As for your "also true" after-question. Consider the case where all the $X_n$ are $N(0,1)$, say. Then for every $M>0$, $\lim_{n\to\infty}P(|X_n|>M) = 2Q(M)>0$, where $Q$ is the gaussian upper-tail integral. Can you check criteria (1) or (2) for $X_n=O_p(1)$?