I have to apply Azouma inequality to a Doob martingale. To do this I have to prove that my Doob martingale has bounded increments.

Wikipedia explains a tecnique which could be very useful to me in order to bound increments (see Doob martingale - Wikipedia, the free encyclopedia "McDiarmid's Inequality" paragraph),

but I can't understand a passage: why if the function f has bounded increments, then the martingale B has bounded increments?

Could someone help me to write down a proof?