Hey, I've been having trouble with these two inequalities. I did the basis step but I'm having trouble proving for n+1.

1) for n = 1, 2.. and a(i) are positive numbers

$\displaystyle (a_{1}*a_{2}...*a_{2^n})^\frac{1}{2^n} \leq \frac{a_{1} + a_{2} + ... + a_{2^n}}{2^n}$

revised (cartesian product cardinality = cardinality of each set multiplied)

2) $\displaystyle \mid X_{1} \times X_{2} ... \times X_{n} \mid = \mid X_{1} \mid * \mid X_{2} \mid ... \mid X_{n} \mid$

Could you please explain how you arrive to each conclusion? Thanks in advance.