Is My Proof on Images and Monotone Functions Correct?

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Homework Statement



Included in the pdf


Homework Equations



Included in the pdf


The Attempt at a Solution



Please take a look at pdf for my attempted solution. I would like to know how if it makes logical sense or if I should change something. Thanks!
 

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Can anyone look at my proof on the attached pdf and tell me if it is correct or on the right track?
 

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