How to turn convergence in probability a statement involving n?

Question

Def: for every $\epsilon$ $\lim_{n}P(|X_n-X|>\epsilon)=0$

How to turn it into a statement of there is an N s.t. n>N...

Shall we make it $P(\lim_{n}|X_n-X|>\epsilon)=0$ first?

Answer 1

For every $\epsilon>0,\delta>0$, there exist $N$,such that $P(|X_n-X|>\epsilon)<\delta$, for all $n\ge N$.

Actually, you can verify it's the same as for every $\epsilon>0$, there exist $N$,such that $P(|X_n-X|>\epsilon)<\epsilon$, for all $n\ge N$.