<s>[INST] Prove the function Prove 
$\forall n\in\mathbb{N}: n\ge 1 \rightarrow 2^n\le 2^{n+1}-2^{n-1}-1.$
I did that $2^n \le\ 2^{n+2} - 2^{n}$ and then $2^n < 2^{n+2} - 2^{n-1}$
but have no idea how to add $-1$ in the function and let the $2^n \le\ 2^{n+2} - 2^{n} - 1$
 [/INST] $2^{n+1}-2^{n-1}-1=2^{n-1}*(4-1)-1=2^{n-1}*3 -1 \ge 2^{n-1}*3-2^{n-1}=2^{n-1}*2=2^n $
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