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**Opalg** Integrating by parts as previously, we get $\displaystyle \int_{x_0}^x\!\!\! f(t)g(t)\,dt = F(x)g(x) - F(x_0)g(x_0) - \int_{x_0}^x\!\!\! F(t)g'(t)\,dt$. Use the fact that $\displaystyle g(x) = g(x_0) + \int_{x_0}^x\!\!\! g'(t)\,dt$ to write this as $\displaystyle \int_{x_0}^x \!\!\! f(t)g(t)\,dt = F(x)g(x_0) - F(x_0)g(x_0) + \int_{x_0}^x \bigl(F(x)-F(t)\bigr)g'(t)\,dt$. The first two terms on the right side of that equation are bounded, so when we divide by g(x) and let $\displaystyle x\nearrow b$ they will go to 0. Thus we need only estimate the size of the integral term.