
Integration Theorem
I have this in my notes:
Define continuous functions on [a,b] by and Then and . The result follows from linearity and the triangle inequality by the following calculation:
.
I understand that but not why .
Any help in this will be much appreciated thank you :)

It uses the fact that .

What I have written is the proof for that fact but its one of the steps that I don't understand :(

Oh, of course, I completely misread.
Do you see that if f(x) is nonnegative on (a,b) then ?
Thats true because both f(x) and the integral are never negative so you can just "discard" the absolute values.
Now, by its definition, is nonnegative. By its definition, is nonpositive so is is nonnegative.
Clearly, while . If that's not clear, think about what happens for each x if (1) , (2) .
Now, while .
Finally, use the fact that, for numbers x and y, .
