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I'm following the proof in this video (https://www.youtube.com/watch?v=lY2WG7xM76I&t=195s), but I am confused with one of the steps. (3:21)

Why is the following true?

$$[X_n \leq x] + [X_n>x,-X \geq -x + \varepsilon] \leq [X_n \leq x]+ [X_n-X \geq \varepsilon] $$

Or in simplier terms why is the following true?

$$[X_n>x,-X \geq -x + \varepsilon] \leq [X_n-X \geq \varepsilon] $$

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  • $\begingroup$ what video${}$? $\endgroup$ Dec 2, 2019 at 4:43
  • $\begingroup$ Apologies - added the link. $\endgroup$
    – Eisen
    Dec 2, 2019 at 4:45
  • $\begingroup$ It is actually $[X_n > x, -X \geq -x+\epsilon]$ in the second term LHS, thats why it works. $\endgroup$
    – fGDu94
    Dec 2, 2019 at 5:31
  • $\begingroup$ Thanks I edited my post. However I still can't see how this is larger because the inequalities are different on both sides. $\endgroup$
    – Eisen
    Dec 2, 2019 at 13:22
  • $\begingroup$ See also en.m.wikipedia.org/wiki/… $\endgroup$ Dec 2, 2019 at 13:32

1 Answer 1

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You can rewrite $[X_n>x,-X\geq -x+\varepsilon]$ as $[X_n>x,X\leq x-\varepsilon]$.

If $X_n>x$ and $X\leq x-\varepsilon$ then also $X_n-X>\varepsilon$. Since $X_n-X>\varepsilon$ occurs whenever $X_n>x,X\leq x-\varepsilon$ occurs, its probability is at least as high.

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