I have to show the following by induction.
$1 \cdot 2 \cdot 3 ... (n - 1) \leq (\frac{n}{2})^{n -1}$
As it is homework I "only" need a push in the right direction. my thought is that is something to do with the binomial theorem.. but I'm pretty lost.
\begin{align*} 1&\le\left(\frac12\right)^0\\ 1&\le1\\ 1\cdot2\cdot3\cdot\ldots\cdot(k-1)&\le\left(\frac k2\right)^{k-1}\\ k!&\le\left(\frac{k+1}2\right)^k\\ \left(\frac k2\right)^{k-1}k&\le\left(\frac{k+1}2\right)^k\\ 2\cdot k^k&\le(k+1)^k \end{align*}
Fourth row is what needs to be proven and 5, 6 are what is sufficient to prove. I guess you could take it from here.