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On pose : <math>\displaystyle{\sup_{\mathcal{R}}sup\Big([A,{(A_i)}_{i \in I}]\Big) = \sup_{\mathcal{R}}sup \Big(\lim_{i \in I, \,\, i \rightarrow \sup_{classique,\mathcal{R}}(I)} A_i\Big) \,\, \underset{d\acute{e}f}{=} \,\, \lim_{i \in I, \,\, i \rightarrow \sup_{classique,\mathcal{R}}(I)} \sup_{classique,\mathcal{R}}(A_i)}</math>.
Si, de plus, <math>[A,{(A_i)}_{i \in I}]</math> est un plafonnement normal de la partie <math>A</math>,
alors on pose : <math>\displaystyle{\sup_{\mathcal{R}}sup(A) \,\, \underset{d\acute{e}f}{=} \,\, \sup_{normale,\mathcal{R}}(A) \,\, \underset{d\acute{e}f}{=} \,\, \sup_{\mathcal{R}}sup\Big([A,{(A_i)}_{i \in I}]\Big)}</math>
et il ne dépend pas du représentant du plafonnement normal de la partie <math>A</math> choisi.
On pose : <math>\lim_{i \in I, \,\, i \rightarrow \sup_{\mathcal{R}}sup(I)} A_i \,\, \underset{d\acute{e}f}{=} \,\, \lim_{i \in I, \,\, i \rightarrow \sup_{classique,\mathcal{R}}(I)} A_i</math>
On a :
<math>\displaystyle{\sup_{\mathcal{R}}sup\bigg(\Big[2\N,{\Big(\N \bigcap [0,2n]\Big)}_{n \in \N}\Big]\bigg) = 2 {card}_{Q,\mathcal{R}}\bigg(\Big[\N,{\Big(\N \bigcap [0,n]\Big)}_{n \in \N}\Big]\setminus \{0\}\bigg) = 2 \sup_{\mathcal{R}}sup\bigg(\Big[\N,{\Big(\N \bigcap [0,n]\Big)}_{n \in \N}\Big]\bigg) \in +\infty}</math>
<math>\displaystyle{\sup_{\mathcal{R}}sup\bigg(\Big[2\N+1,{\Big(\N \bigcap [1,2n+1]\Big)}_{n \in \N}\Big]\bigg) = 2 {card}_{Q,\mathcal{R}}\bigg(\Big[\N,{\Big(\N \bigcap [0,n]\Big)}_{n \in \N}\Big]\setminus \{0\}\bigg) + 1 = 2 \sup_{\mathcal{R}}sup\bigg(\Big[\N,{\Big(\N \bigcap [0,n]\Big)}_{n \in \N}\Big]\bigg) + 1 \in +\infty}</math>
On pose :
<math>\displaystyle{\sup_{\mathcal{R}}sup(\N) \,\, \underset{d\acute{e}f}{=} \,\, \sup_{normale,\mathcal{R}}(\N) \,\, \underset{d\acute{e}f}{=} \,\, \sup_{\mathcal{R}}sup\bigg(\Big[\N,{\Big(\N \bigcap [0,n]\Big)}_{n \in \N}\Big]\bigg) \in +\infty}</math>,
<math>\displaystyle{\sup_{\mathcal{R}}sup(2\N) \,\, \underset{d\acute{e}f}{=} \,\, \sup_{normale,\mathcal{R}}(2\N) \,\, \underset{d\acute{e}f}{=} \,\, \sup_{\mathcal{R}}sup\bigg(\Big[2\N,{\Big(\N \bigcap [0,2n]\Big)}_{n \in \N}\Big]\bigg) \in +\infty}</math>,
<math>\displaystyle{\sup_{\mathcal{R}}sup(2\N+1) \,\, \underset{d\acute{e}f}{=} \,\, \sup_{normale,\mathcal{R}}(2\N+1) \,\, \underset{d\acute{e}f}{=} \,\, \sup_{\mathcal{R}}sup\bigg(\Big[2\N+1,{\Big(\N \bigcap [1,2n+1]\Big)}_{n \in \N}\Big]\bigg) \in +\infty}</math>.
On a : <math>\sup_{\mathcal{R}}sup(2\N+1) = 2 \sup_{\mathcal{R}}sup(\N) + 1 > 2 \sup_{\mathcal{R}}sup(\N) = \sup_{\mathcal{R}}sup(2\N) > \sup_{\mathcal{R}}sup(\N)</math>.
Soit <math>f \in \mathcal{F}(\N, \R_+ \bigsqcup +\infty), \,\, strict. \,\, \nearrow</math>
et telle que <math>\displaystyle{\lim_{n \in \N, \,\, n \rightarrow \sup_{\mathcal{R}}sup(\N)} f(n) \in +\infty}</math>
(où <math>\displaystyle{\lim_{n \in \N, \,\, n \rightarrow \sup_{\mathcal{R}}sup(\N)} f(n) \in +\infty \,\, \underset{d\acute{e}f}{\Leftrightarrow} \,\, \lim_{n \in \N, \,\, n \rightarrow \sup_{classique,\mathcal{R}}(\N)} f(n) \in +\infty \,\, \underset{d\acute{e}f}{\Leftrightarrow} \,\,\underset{n \in \N, \,\, n \rightarrow \sup_{classique,\mathcal{R}}(\N)}{\text{lim}_{classique}} f(n) = +\infty_{classique}}</math>)
et telle que <math>\displaystyle{\lim_{n \in \N, n \rightarrow \sup_{\mathcal{R}}sup(\N)} f(n) = f(\lim_{n \in \N, n \rightarrow \sup_{\mathcal{R}}sup(\N)} n)}</math>.
<math>\displaystyle{\sup_{\mathcal{R}}sup\Big(f(\N)\Big) \,\, \underset{d\acute{e}f}{=} \,\, \sup_{normale,\mathcal{R}}\Big(f(\N)\Big) \,\, \underset{d\acute{e}f}{=} \,\,\sup_{\mathcal{R}}sup\bigg(\Big[f(\N),{\Big(\N \bigcap [f(0),f(n)]\Big)}_{n \in \N}\Big]\bigg) = f\Bigg({card}_{Q,\mathcal{R}} \bigg(\Big[\N,{\Big(\N \bigcap [0,n]\Big)}_{n \in \N}\Big]\setminus \{0\}\bigg)\Bigg) = f \Bigg(\sup_{\mathcal{R}}sup\bigg(\Big[\N,{\Big(\N \bigcap [0,n]\Big)}_{n \in \N}\Big]\bigg)\Bigg)}</math>
<math>\displaystyle{= f\Big(\sup_{normale,\mathcal{R}}(\N)\Big) = f\Big(\sup_{\mathcal{R}}sup(\N)\Big) \in +\infty}</math>.
'''a)'''
"<math>\displaystyle{\lim_{i \in I, \,\, i \rightarrow \sup_{\mathcal{R}}sup(I)} {card}_{Q,\mathcal{R}} (A_i)}</math>"
ou "<math>\displaystyle{\lim_{i \in I, \,\, i \rightarrow \sup_{classique,\mathcal{R}}(I)} {card}_{Q,\mathcal{R}} (A_i)}</math>"
ou "<math>\displaystyle{{\underset{classique}{\lim}}_{i \in I, \,\, i \rightarrow \sup_{classique,\mathcal{R}}(I)} {card}_{Q,\mathcal{R}} (A_i)}</math>".
'''b)'''
"<math>\displaystyle{\lim_{i \in I, \,\, i \rightarrow \sup_{\mathcal{R}}sup(I)} \sup_{\mathcal{R}}sup(A_i)}</math>"
ou "<math>\displaystyle{\lim_{i \in I, \,\, i \rightarrow \sup_{classique,\mathcal{R}}(I)} \sup_{\mathcal{R}}sup(A_i)}</math>"
ou "<math>\displaystyle{\lim_{i \in I, \,\, i \rightarrow \sup_{classique,\mathcal{R}}(I)} \sup_{classique,\mathcal{R}}(A_i)}</math>"
ou "<math>\displaystyle{{\underset{classique}{\lim}}_{i \in I, \,\, i \rightarrow \sup_{classique,\mathcal{R}}(I)} \sup_{\mathcal{R}}sup(A_i)}</math>"
ou "<math>\displaystyle{{\underset{classique}{\lim}}_{i \in I, \,\, i \rightarrow \sup_{classique,\mathcal{R}}(I)} \sup_{classique,\mathcal{R}}(A_i)}</math>".
L'application borne supérieure sur <math>\mathbb{R}^{n}</math>, relative au repère orthonormé <math>\mathcal{R}</math>, <math>\sup_{\mathcal{R}}sup</math>,
est une application :
<math>\displaystyle{\sup_{\mathcal{R}}sup \,\, : \,\, \mathcal{P}(\R^n) \bigsqcup {\mathcal{P}lafonnements}\Big(I,\mathcal{P}(\R^n)\Big) \,\, \longrightarrow \,\, -\infty \bigsqcup \R \bigsqcup +\infty}</math>,
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