I was reading the proof of pathwise uniqueness in the book of Ikeda and Watanabe but I am not sure where is the boundedness of $\sigma(x)$ and $b(x)$ being used? I am attaching the proof below and I think the boundedness is being used at the place marked in red since boundedness ensures thatthe stochastic integral and the integral in red are well defined. Am I correct or is it being used somewhere else?
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