$\textbf{I would like a proof in detail of the following question.}$ 

let $ f:[a,b)\rightarrow \mathbb{R} $ be a bounded function which is Riemann integrable on [a,c] whenever $a< c< b$. Define the function $F:\left[ a,b\right)\rightarrow \mathbb{R} $ by the formula $F(x)=\int_{a}^{x} f$Prove that $F$ has a limit at $b$ and the integral of $f$ over $[a,b)$ can be defined to equal this limit.


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