Question in Understanding "converges in probability"

The definition says "{Xn} converges in probability to X iff P(|Xn - X|
= ε) ---> 0 for every ε > 0". I am wondering if it can be defined
without introducing ε? Specifically, what's the difference between "P(|
Xn - X| > ε) ---> 0 for every ε > 0" and "P(|Xn - X| > 0) ---> 0"? If
they are not the same, as a condition which one is stronger ? Please
give some clarification. Thanks in advance!
.

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