# Convergence in probability - equivalence of alternative fomulations

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.

• This is not really about convergence in probability but about different formulations of the limit of a sequence. – Calculon Oct 18 '18 at 20:01

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 $$\lim_{n \to \infty}P(\vert X_n -X \vert > \epsilon)=0$$