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I'm trying to prove that $n! > n^2$ for $n\geq 4$ by use of mathematical induction, but I get to the inductive step and get lost. But I'm struggling with the inductive step as expected.

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marked as duplicate by Martin Sleziak, Aloizio Macedo, graydad, Jyrki Lahtonen Oct 4 '15 at 16:26

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$4!=24>16=4^2$, so the basic step holds. Then:

$$ (n+1)! = (n+1) n! \color{red}{>} (n+1) n^2 > (n+1)(n+1) = (n+1)^2 $$ and we are fine. We used the inductive hypothesis in $\color{red}{>}$.

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HINT: multiplying $$n!>n^2$$ by $n+1>0$ we get $$(n+1)!>n^2(n+1)$$ and now you have to show that $$n^2(n+1)>(n+1)^2$$ which is easy.

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  • $\begingroup$ You probably start from $$n!>n^2,$$ not $$(n+1)!>(n+1)^2.$$ $\endgroup$ – Did Oct 3 '15 at 14:35
  • $\begingroup$ this step is trivial i have it not forgotten, what is the matter with you? $\endgroup$ – Dr. Sonnhard Graubner Oct 3 '15 at 14:36
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    $\begingroup$ The matter with you is that you cannot even set correctly a proof by induction (and we will not mention initialization so that kids won't be afraid...). $\endgroup$ – Did Oct 3 '15 at 14:40
  • $\begingroup$ yes you have ritgh, i have made a typo, sorry for it! $\endgroup$ – Dr. Sonnhard Graubner Oct 3 '15 at 14:40

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