Chebyschev’s inequality

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Can someone please help me understand steps 4 and 5 of this proof?

What I think is that the integral is being split into the sum of the inside of the region and the outside, but then there's also step 5 where the (x-mu)^2 of the second integral is reduced to 0 and I don't quite understand why?

Any help is appreciated!

[quiet sparrow]

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[safe flint]

Well in the fourth step for any x (x-u)^2>=0 because it’s always positive and for the first integral you integrate on the intervals for which |x-u|>epsilon thus (x-u)^2> epsilon^2 in that case via continuity of the integral you have the result on line 4 and 5

Here « u » denotes mu

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Okay thank you, I just had my aha moment. Sometimes you just need the extra push.

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Appreciate it