For all $n \in \mathbb { N } ^ { * }$, consider in $]0,1[$ the points $a _ { k , n } = \frac { 2 k + 1 } { 2 n }$ for $k \in \llbracket 0 , n - 1 \rrbracket$, and $S _ { n } \left( h _ { \alpha } \right) = \frac { 1 } { n } \sum _ { k = 0 } ^ { n - 1 } h _ { \alpha } \left( a _ { k , n } \right)$, where $J_\alpha = \int_0^1 h_\alpha(t)\,\mathrm{d}t$. Deduce that the sequence $\left( S _ { n } \left( h _ { \alpha } \right) \right) _ { n \in \mathbb { N } ^ { * } }$ converges to $J _ { \alpha }$.
For all $n \in \mathbb { N } ^ { * }$, consider in $]0,1[$ the points $a _ { k , n } = \frac { 2 k + 1 } { 2 n }$ for $k \in \llbracket 0 , n - 1 \rrbracket$, and $S _ { n } \left( h _ { \alpha } \right) = \frac { 1 } { n } \sum _ { k = 0 } ^ { n - 1 } h _ { \alpha } \left( a _ { k , n } \right)$, where $J_\alpha = \int_0^1 h_\alpha(t)\,\mathrm{d}t$. Deduce that the sequence $\left( S _ { n } \left( h _ { \alpha } \right) \right) _ { n \in \mathbb { N } ^ { * } }$ converges to $J _ { \alpha }$.