I was able to prove Bernoulli's inequality, easily by simple induction.
However, I'm not sure how to prove the generalized inequality (generalized = for each sequence of numbers $i=1,\ldots,n$):
$$\prod\limits_{i = 1}^n {(1 + {x_i})} \ge 1 + \sum\limits_{i = 1}^n {{x_i}},\qquad {x_i} \ge 0$$
How do you prove it? Thanks!