(All Steps) Prove that the following are equivalent. X_n \rightarrow X in probability. There exist ε_n → 0 such that P(|X_n-X|>ε_n)
Question: Prove that the following are equivalent.
- \(X_{n} \rightarrow X\) in probability.
- There exist \(\varepsilon_{n} \downarrow 0\) such that \(P\left(\left|X_{n}-X\right|>\varepsilon_{n}\right) \leq \varepsilon_{n}\)
- \(E \min \left(\left|X_{n}-X\right|, 1\right) \rightarrow 0\)
Deliverable: Word Document 
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