Question Number 198077 by SANOGO last updated on 10/Oct/23
Answered by Mathspace last updated on 10/Oct/23
$${par}\:{recurrence}\:{sur}\:{n} \\ $$$${on}\:\Phi\left(\mathrm{0}\right)\geqslant\mathrm{0}\:{vraie}\:{puisque}\:\Phi\left({N}\right)\subset{N} \\ $$$${supposons}\:{que}\:\Phi\left({n}\right)\geqslant{n}\:{et}\:{montrons} \\ $$$${que}\:\Phi\left({n}+\mathrm{1}\right)\geqslant{n}+\mathrm{1} \\ $$$${on}\:{n}+\mathrm{1}>{n}\:{et}\:\Phi\:{strictement}\: \\ $$$${croissante}\:\Rightarrow\Phi\left({n}+\mathrm{1}\right)>\Phi\left({n}\right)\geqslant{n} \\ $$$$\left({hypothese}\:{de}\:{recurrence}\right)\Rightarrow \\ $$$$\Phi\left({n}+\mathrm{1}\right)>{n}\:{cad}\:\Phi\left({n}+\mathrm{1}\right)\geqslant{n}+\mathrm{1} \\ $$$${la}\:{relation}\:{est}\:{vraie}\:{a}\:{l}\:{ordre} \\ $$$${n}+\mathrm{1} \\ $$
Commented by SANOGO last updated on 10/Oct/23
$${merci} \\ $$