## total variation problem

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

there are some lemmas, etc that got me this far:
$$\int_{a}^{b} f' \leq f(b)-f(a) = P_a^b - N_a^b \leq P_a^b + N_a^b = T_a^b f$$ 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:
$$\vert \int_E f \vert \leq \int_E |f|$$
would that help at all?

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 up... can anyone help?