I'm having trouble understanding the answer to one of the mit opencourseware Statistic For Applications homeworks
The question: (question 1.): [0] The answer: [1]
The question asks to show this random variable converges in probability:
$P[X_n=1/n] = 1-1/n^2$
$P[X_n=n] = 1/n^2$
The answer first calculates the expected value:
$E[x_n]=2/n-1/n^3$
Then it says "On the other hand:
$\lim_{n \to inf} P[|X_n|>\epsilon] = \lim_{n \to inf} P[X_n>\epsilon] <= (\lim_{n \to inf}E[X_n])/\epsilon = \lim_{n \to inf}(2/n-1/n^3)/\epsilon = 0$
...hence Hence $X_n$ converges in probability"
The part I don't understand is this:
$\lim_{n \to inf} P[X_n>\epsilon] <= (\lim_{n \to inf}E[X_n])/\epsilon$
Where did that come from? I've been looking online to see if I can find any examples that does something similar but I have yet to see any. As far as I can tell the expected value shouldn't have this relationship to the probability of the value of $X_n$. Is this a common tick to use to solve these problems? Can someone give me some intuition as to how this "<=" makes sense?
[1] https://github.com/hoangnguyen7699/StatisticsForApplication_solution/blob/master/PS1/ProblemSet1.pdf
