The following text is from my textbook that I have a hard time understanding.
Let $X_1, X_2,X_3,...$ be independent random viariables such that
$P(X_n =1) = 1 - \frac{1}{n} $ and $P(X_n = n) = \frac{1}{n}, 2 \leq n$
Clearly,
$P(|X_n -1| > \epsilon) = P(X_n = n) = \frac{1}{n} \to 0$ as n $\to \infty$
That is $X_n \to 1 $ as $ n \to \infty$ in probability.
My question is why does the following hold?
$P(|X_n -1| > \epsilon) = P(X_n = n)$?
If $|X_n-1|> \epsilon$ then $X_n \neq 1$ so $X_n =n$. If $X_n=n$ then $|X_n-1|=n-1>\epsilon$ provided $n >1+\epsilon$. So $P\{|X_n-1|> \epsilon\}=P\{X_n=n\}$ provided $n >1+\epsilon$.