total variation problem

fourier jr

problem: Let [tex]f \in BV[a,b] [/tex]. Then [tex]\int_{a}^{b} |f'| \leq T_a^b f [/tex] where [tex]T_a^b f [/tex] is the total variation of f over [a,b].

there are some lemmas, etc that got me this far:
[tex]\int_{a}^{b} f' \leq f(b)-f(a) = P_a^b - N_a^b \leq P_a^b + N_a^b = T_a^b f [/tex] where P is the positive variation & N is the negative variation of f.

the absolute value there messes me up; i don't know what to do about it. i know there's a theorem that says the following:
[tex]\vert \int_E f \vert \leq \int_E |f| [/tex]
would that help at all?

fourier jr

up... can anyone help?