Ex2-19

Let {\Gamma_{1},\Gamma_{2}} be partitions of {[a^{\prime},b^{\prime}]} and choose two families of intermediate points for {\Gamma_{1},\Gamma_{2}} respectively. Also, let {\Gamma, \Gamma^{\prime}} be partition of {[a, a^{\prime}]} and {[b^{\prime}, b]}. Then {U=\Gamma_{1}\cup\Gamma\cup\Gamma{\prime}} and {V=\Gamma_{2}\cup\Gamma\cup\Gamma^{\prime}} are two partitions of {[a, b]}. Choose two more families of intermediate points for partitions {\Gamma, \Gamma^{\prime}} and let the union of intermediate points of {\Gamma_{1},\Gamma_{2},\Gamma,\Gamma^{\prime}} be the family of intermediate points of {U} and {V}. We have {R_{U}[a, b]=R_{\Gamma}[a, a^{\prime}]+R_{\Gamma_{1}}[a^{\prime},b^{\prime}]+R_{\Gamma^{\prime}}[b^{\prime},b]} and {R_{V}[a, b]=R_{\Gamma}[a, a^{\prime}]+R_{\Gamma_{2}}[a^{\prime},b^{\prime}]+R_{\Gamma^{\prime}}[b^{\prime},b]}. Now, given any {\epsilon>0} such that {|R_{\Gamma_{1}}[a^{\prime},b^{\prime}]-R_{\Gamma_{2}}[a^{\prime},b^{\prime}]|<\epsilon}, since {\int_a^b \! f \, \mathrm{d}\phi} exists, there is some {\delta>0} such that {|U|,|V|<\delta} implies
{|R_{\Gamma_{U}}[a,b]-R_{\Gamma_{V}}[a,b]|<\epsilon}. Finally, assuming {|\Gamma|,|\Gamma^{\prime}|<\delta} and suppose that {|\Gamma_{1}|,|\Gamma_{2}|<\delta}, we have {|U|=|\Gamma_{1}\cup\Gamma\cup\Gamma{\prime}|, |V|=|\Gamma_{2}\cup\Gamma\cup\Gamma{\prime}|<\delta}, this implies {|R_{\Gamma_{U}}[a,b]-R_{\Gamma_{V}}[a,b]|=|R_{\Gamma_{1}}[a^{\prime},b^{\prime}]-R_{\Gamma_{2}}[a^{\prime},b^{\prime}]|<\epsilon}, which complete the proof