Someone has asked similiar question before, but the answer to that question omits the proof. And I couldn't find the proof on Rudin's Real and Complex Analysis or Stein's Real Analysis.
Prove that :$ \int_a^b |f'(x)|\,dx \leq V(f,[a,b])$, where f is a function of bounded variation.
I tried to imitate the proof of "$ \int_a^b |g'(x)|\,dx \ = V(g,[a,b])$, where g is absolutely continuous", but failed. I also tried to use the Lebsgue decomposition, but I didn't know what to do next.
Any hint will be appreciated!!