When you $\displaystyle \int_0^1$ cos[n(x+1)] as n->$\displaystyle \infty$, at one point you say lim n->$\displaystyle \infty$ 1/n [sin2n -sin(n)] $\displaystyle \leq$ lim n->$\displaystyle \infty$ 2/n = 0 .

Why is it okay to say $\displaystyle \leq$ here? How do you know that the integral is 0, and not something less than 0.