ليكن $\displaystyle{\displaylines{n \in \mathbb{N}^{*}}}$ و $\displaystyle{\displaylines{a_{1},\, a_{2},\, .... a_{n}}}$ أعداد حقيقية موجبة قطعاً .
حدد النهاية التالية :
$\displaystyle{\displaylines{\lim_{p\rightarrow +\infty} \left( \sum_{k=1}^{n}a_{k}^{p} \right)^{\frac{1}{p}}}}$
ليكن $\displaystyle{\displaylines{m = \max_{1\le k \le n} a_{k}}}$, بحيث $\displaystyle{\displaylines{m = a_{i} \, , \, i \in \{1,\cdots, n\}}}$
لدينا $\displaystyle{\displaylines{\forall k \in \{1, 2, ..., n\}\,:\, a_{k} \le \max_{1\le j \le n} a_{j} = m}}$
إذن $\displaystyle{\displaylines{(\forall p \in \mathbb{N}^{*}) \ (\forall k \in \{1, 2, ..., n\}) \,:\, a_{k}^{p} \le m^{p}}}$
نقوم بالجمع : $\displaystyle{\displaylines{\sum_{k=1}^{n}a_{k}^{p} \le n m^{p}}}$
$\displaystyle{\displaylines{\forall p \in \mathbb{N}^{*} \ : \ \left( \sum_{k=1}^{n}a_{k}^{p} \right)^{\frac{1}{p}} \le n^{\frac{1}{p}} m \tag{1}}}$
و من جهة أخرى لدينا : $\displaystyle{\displaylines{\sum_{k=1}^{n}a_{k}^{p} = m^{p} + \sum_{\substack{k=1 \\ k \neq i}}^{n}a_{k}^{p}}}$
إذن $\displaystyle{\displaylines{m^{p} \le \sum_{k=1}^{n}a_{k}^{p}}}$
$\displaystyle{\displaylines{\forall p \in \mathbb{N}^{*} \ : \ m \le \left( \sum_{k=1}^{n}a_{k}^{p} \right)^{\frac{1}{p}} \tag{2}}}$
من $\displaystyle{\displaylines{(1)}}$ و $\displaystyle{\displaylines{(2)}}$ لدينا : $\displaystyle{\displaylines{\forall p \in \mathbb{N}^{*} \ : \ m \le \left( \sum_{k=1}^{n}a_{k}^{p} \right)^{\frac{1}{p}} \le n^{\frac{1}{p}} m}}$
بما أن : $\displaystyle{\displaylines{ \lim_{p\rightarrow +\infty} n^{\frac{1}{p}} m = \lim_{p\rightarrow +\infty} m = \max_{1\le k \le n} a_{k} = a_{i}}}$
فإن : $\displaystyle{\displaylines{\lim_{p\rightarrow +\infty} \left( \sum_{k=1}^{n}a_{k}^{p} \right)^{\frac{1}{p}} = \max_{1\le k \le n} a_{k}}}$
