For all $n \in \mathbb { N } ^ { * }$, consider in $]0,1[$ the points $a _ { k , n }$ given, for $k \in \llbracket 0 , n - 1 \rrbracket$, by $a _ { k , n } = \frac { 2 k + 1 } { 2 n }$ and $$S _ { n } \left( h _ { \alpha } \right) = \frac { 1 } { n } \sum _ { k = 0 } ^ { n - 1 } h _ { \alpha } \left( a _ { k , n } \right).$$ For all $n \in \mathbb { N } ^ { * }$, show that $$\int _ { 1 / 2 n } ^ { ( 2 n - 1 ) / 2 n } h _ { \alpha } ( t ) \, \mathrm { d } t + \frac { 1 } { n } h _ { \alpha } \left( \frac { 2 n - 1 } { 2 n } \right) \leqslant S _ { n } \left( h _ { \alpha } \right) \leqslant \frac { 1 } { n } h _ { \alpha } \left( \frac { 1 } { 2 n } \right) + \int _ { 1 / 2 n } ^ { ( 2 n - 1 ) / 2 n } h _ { \alpha } ( t ) \, \mathrm { d } t.$$
For all $n \in \mathbb { N } ^ { * }$, consider in $]0,1[$ the points $a _ { k , n }$ given, for $k \in \llbracket 0 , n - 1 \rrbracket$, by $a _ { k , n } = \frac { 2 k + 1 } { 2 n }$ and
$$S _ { n } \left( h _ { \alpha } \right) = \frac { 1 } { n } \sum _ { k = 0 } ^ { n - 1 } h _ { \alpha } \left( a _ { k , n } \right).$$
For all $n \in \mathbb { N } ^ { * }$, show that
$$\int _ { 1 / 2 n } ^ { ( 2 n - 1 ) / 2 n } h _ { \alpha } ( t ) \, \mathrm { d } t + \frac { 1 } { n } h _ { \alpha } \left( \frac { 2 n - 1 } { 2 n } \right) \leqslant S _ { n } \left( h _ { \alpha } \right) \leqslant \frac { 1 } { n } h _ { \alpha } \left( \frac { 1 } { 2 n } \right) + \int _ { 1 / 2 n } ^ { ( 2 n - 1 ) / 2 n } h _ { \alpha } ( t ) \, \mathrm { d } t.$$