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Why are the following two statements about convergence in probability the same. Probably a silly question, but I don't see the direct link

$$P(|X_n| > \delta) \rightarrow 0$$

vs.

$$P(|X_n| > M) < \epsilon $$

for sufficiently large $n$, where $\epsilon$, $\delta$ and $M$ are some positive constants.

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    $\begingroup$ This is not really about convergence in probability but about different formulations of the limit of a sequence. $\endgroup$ – Calculon Oct 18 '18 at 20:01
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The definition of convergence in probability should be:

A sequence of random variables $X_n$ is said to converge to a random variable $X$ in probability if for $\mathbf {every}$ $\epsilon >0$ we have $$\lim_{n \to \infty}P(\vert X_n -X \vert > \epsilon)=0$$

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