Hello I have been blasting at this inequality proof and it is just not doing what I want it to do:
Prove that $2^{n-1}(a_1a_2\ldots a_n + 1) \geq (1+a_1)(1+a_2)\ldots(1+a_n)$ assuming that $a_1,a_2,\dots, a_n \geq 1$.$\:\:\:$
So the base case is pretty trivial, which leads me to assume that this inequality holds for $k$, so I want to prove $P(k+1)$. I do the following:
$$ (1+a_1)(1+a_2)\ldots(1+a_k)(1+a_{k+1})\leq 2^{k-1}(a_1a_2\ldots a_k + 1)(a_{k+1} + 1) $$ I tried methods like setting the second "$1$" to $a_{k+1}$ and then I have $2^k(a_1\ldots a_k + 1)a_{k+1}$ but I can't assume from there that $2^k(a_1...a_ka_{k+1} +a_{k+1})$ and the second $a_{k+1}$ becomes $1$. So maybe my method is not good from the get-go. Thank you!