i have a question from an analysis paper that i think can be solved using the arithmetic-geometric means inequality. This is the question:

show that if a_1, a_2, a_3, ... , a_n are positive, such that

(a_1)(a_2)(a_3)...(a_n)=1

then

(1+a_1) (1+a_2) (1+a_3) ... (1+a_n) ≥ 2^n

it makes sense because using the AM-GM inequality, 'on average' the a_i are greater than 1, so 'on average you have something greater than

(1+1) (1+1) ...(1+1) = 2^n

but i don't know how to prove it rigourously.

can anyone help?