I am doing some exercises for my real analysis class and I ran into one that has been bugging me all day. I am using the book Elementary Analysis Theory of Calculus by Kenneth Ross. The problem is in Section 33 and is exercise 33.6 and it reads the following:
Prove that for any subset S of [a,b], M(|f|,S)-m(|f|,S)<=M(f,S)-m(f,S). Hint: For x,y in S |f(x)|-|f(y)|<=|f(x)-f(y)|<=M(f,s)-m(f,s).
Note that this book might have different notation than what some of you are used to but just to be clear, M(f,S)=sup{f(x):x in S} and m(f,S)=inf{f(x):x in S}.
Any help will be appreciated. Thanks..