Please tell me if the following is sufficient to prove that lim sup (sin x) = 1. I have marked steps that I might need to explain--i.e., prove as well--with a **. My question is, at an Analysis level, would this proof suffice?
Part A: Show lim sup (sin x) >= 1.
**1. Observe that sin(2*n*pi + x) = sin x for any integer n.
2. sin (pi/2) = 1.
3. x = pi/2 => sin(2*n*pi + pi/2) = 1 for any integer n.
4. => lim sup (sin x) >= 1.
Part B: Show lim sup (sin x) >= 1.
**5. The domain of sin x is [-1, 1].
6. => lim sup (sin x) <= 1.
7. Combining statements 4 and 6, lim sup (sin x) = 1.
Again, at the Analysis level, would I need to prove steps 1 and 5, or is what I have up here sufficient?