I'm confused on the procedure used to show the theorem 5.52 in van der Vaart "Asymptotic Statistics" p.75. Here the simplified idea. Consider the sequence of real-valued positive random variables $\{X_n\}$ and suppose we want to show that
(*) $X_n \in O_p(1)$ : $\forall K>0$ $\exists M>0$ s.t. $P(X_n>M)<K$ $\forall n \in \mathbb{N}$
Suppose we show that
(**) $\lim_{n \rightarrow \infty} P(X_n>M)=0$ for some $M>0$: $\forall \epsilon>0$ $\exists \bar{n}\in \mathbb{N}$ s.t. $P(X_n>M)<\epsilon$ $\forall n \geq \bar{n}\in \mathbb{N}$
Does (**) imply (*)?
I think it doesn't because
(**) implies that: $\forall \epsilon>0$, $\exists A_{\epsilon}:=\max\{X_1,...,X_{\bar{n}-1}, \epsilon\}$ s.t. $P(X_n>M)<A_{\epsilon}$ $\forall n \in \mathbb{N}$.
However, the procedure used in the proof of the theorem 5.52 in van der Vaart "Asymptotic Statistics" p.75 seems to be based on the idea that (**) implies (*).
Could you help me to clarify this?