Question Number 9790 by tawakalitu last updated on 04/Jan/17
$$\mathrm{prove}\:\mathrm{by}\:\mathrm{mathematical}\:\mathrm{induction} \\ $$$$\mathrm{n}!\:\leqslant\:\mathrm{n}^{\mathrm{n}} \\ $$
Commented by FilupSmith last updated on 04/Jan/17
$${n}=\mathrm{1} \\ $$$$\mathrm{1}!\leqslant\mathrm{1}^{\mathrm{1}} \\ $$$$ \\ $$$${n}=\mathrm{2} \\ $$$$\mathrm{2}!\leqslant\mathrm{2}^{\mathrm{2}} \\ $$$$\mathrm{2}\leqslant\mathrm{4} \\ $$$$ \\ $$$$\left({n}+\mathrm{1}\right)!\leqslant\left({n}+\mathrm{1}\right)^{{n}+\mathrm{1}} \\ $$$$\left({n}+\mathrm{1}\right)\centerdot{n}!\leqslant\left({n}+\mathrm{1}\right)^{{n}+\mathrm{1}} \\ $$$${n}!\leqslant\left({n}+\mathrm{1}\right)^{{n}} \\ $$$$\underset{{n}\:{times}} {\mathrm{1}×\mathrm{2}×...×{n}}\leqslant\underset{{n}\:{times}} {\left({n}+\mathrm{1}\right)×...×\left({n}+\mathrm{1}\right)} \\ $$$$\forall{k}\in\left(\mathrm{1},{n}\right),\:\left({n}+\mathrm{1}\right)>{k} \\ $$
Commented by tawakalitu last updated on 05/Jan/17
$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{sir}.\:\mathrm{God}\:\mathrm{bless}\:\mathrm{you}. \\ $$
Commented by tawakalitu last updated on 05/Jan/17
$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{sir}.\:\mathrm{God}\:\mathrm{bless}\:\mathrm{you}. \\ $$