How to show that :
$$\forall n\in\mathbb N^*,\forall (a_1,\ldots,a_n)\in([1;+\infty[)^n, \prod_{k=1}^{n}(1+a_k)\le2^{n-1}(1+\prod_{k=1}^{n}a_k)$$
I tried by induction but i don't know how to find this.
2026-04-04 15:18:12.1775315892
How to show this inequality between those products?
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Hint. If $a,\,b \geqslant 1$ then $(1+a)(1+b) \leqslant 2(1+ab).$