For all $n \in \mathbb { N } ^ { * }$ and all $k \in \llbracket 0 , n \rrbracket$, we set $x _ { n , k } = - \sqrt { n } + \frac { 2 k } { \sqrt { n } }$. The function $B _ { n }$ is defined as in Q19, and $\varphi ( x ) = \frac { 1 } { \sqrt { 2 \pi } } \mathrm { e } ^ { - x ^ { 2 } / 2 }$. We fix $\varepsilon > 0$ and $\ell \in \mathbb{R}^+$ such that $\varphi(\ell) \leqslant \frac{\varepsilon}{2}$. Show that there exists a natural number $n _ { 1 }$ such that, for all integers $n \geqslant n _ { 1 }$, $$\sup _ { x \in [ 0 , \ell ] } \left| B _ { n } ( x ) - \varphi ( x ) \right| \leqslant \frac { \varepsilon } { 2 }$$
For all $n \in \mathbb { N } ^ { * }$ and all $k \in \llbracket 0 , n \rrbracket$, we set $x _ { n , k } = - \sqrt { n } + \frac { 2 k } { \sqrt { n } }$. The function $B _ { n }$ is defined as in Q19, and $\varphi ( x ) = \frac { 1 } { \sqrt { 2 \pi } } \mathrm { e } ^ { - x ^ { 2 } / 2 }$. We fix $\varepsilon > 0$ and $\ell \in \mathbb{R}^+$ such that $\varphi(\ell) \leqslant \frac{\varepsilon}{2}$.
Show that there exists a natural number $n _ { 1 }$ such that, for all integers $n \geqslant n _ { 1 }$,
$$\sup _ { x \in [ 0 , \ell ] } \left| B _ { n } ( x ) - \varphi ( x ) \right| \leqslant \frac { \varepsilon } { 2 }$$