In the context of the proof of Theorem 4.3, prove that

\[\lim _{n ightarrow \infty}\left|\int_{a}^{b} g_{m}(x) d F(x)-\int_{a}^{b} g(x) d F(x)ight|

for any \(\delta_{\varepsilon}>0\).

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Theorem 4.3 (Helly and Bray). Let {F} be a sequence of distribution functions and let g be a function that is continuous on [a, b] where 181