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 non-negative 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 non-negative. By its definition, is non-positive so is is non-negative.
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, .