Polished A-A and added new lines for broken enumerates.
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@@ -35,7 +35,8 @@
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\begin{enumerate}
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\item[(a)] For each continuous seminorm $\rho$ on $E$, $[f_\alpha - f]_{u, \rho} \to 0$.
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\item[(b)] $\lim_{\alpha \in A}\int_a^b f_\alpha dG$ exists.
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\end{enumerate}
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\end\{enumerate\}
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then $f \in RS([a, b], G)$ and $\int_a^b f dG = \lim_{\alpha \in A}\int_a^b f_\alpha dG$. In particular,
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\begin{enumerate}
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\item If $H$ is complete, then condition (b) may be omitted.
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@@ -55,11 +56,13 @@
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\begin{enumerate}
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\item $[f - f_\alpha]_E < \eps/(3[G]_{\text{var}, F})$.
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\item $\rho\paren{\int_a^b f_\alpha dG - \lim_{\alpha \in A}\int_a^b f_\alpha dG} < \eps/3$.
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\end{enumerate}
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\end\{enumerate\}
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Since $f_\alpha \in RS([a, b], G)$, there exists $P_0 \in \scp([a, b])$ such that if $P \ge P_0$,
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\begin{enumerate}
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\item[(3)] $\rho\paren{S(P, c, f_\alpha, G) - \int_a^b f_\alpha dG} < \eps/3$.
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\end{enumerate}
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\end\{enumerate\}
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Thus for any $(P, c) \in \scp_t([a, b])$ with $P \ge P_0$,
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\[
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\rho\paren{S(P, c, f, G) - \lim_{\alpha \in A}\int_a^b f_\alpha dG} < \eps
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