This is true for all x & y: . Therefore in your case, .
So it follows that .
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?