<s>[INST] Please help me prove by induction that $n^n>1\cdot3\cdot5\cdot\ldots\cdot(2n-1)$ Please help me prove by induction that
$$ 
n^n>1\cdot3\cdot5\cdot\ldots\cdot(2n-1)
$$
 [/INST] We need to prove $1\cdot 3\cdot 5\cdot \ldots \cdot (2n-1)< n^n$
For $n=1$ we have $1 < 1$, what is wrong.
For $n=2$ we have $3 < 4$ what is true.
Note that $1\cdot 3\cdot 5\cdot \ldots \cdot (2n-1) (2\cdot (n+1)-1)=n^n(2(n+1)-1)$ 
then prove $(n+1)^{n+1} - n^n\cdot (2\cdot (n+1)-1) > 0$
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