Would these witnesses satisfy this big-O function?

Question

I'm trying to determine if $f(x) = \lceil x/2 \rceil$ is $O(x)$.

I know that this is true, and the textbook answer is:

$|\lceil x/2\rceil|\leq |(x/2)+1| \leq C|x|$ for all $x > 2$,

with witnesses $C = 1$, $k = 2$.

Would it also be correct if $C = 2$, $k = 0$? Or $C = 1,\ k = 1$? Just want to make sure I'm understanding correctly.

Answer 1

Note that you need to show that $|f(x)| < cx $ for some $c$ for all $x > x_0$ for some $x_0$.

Therefore any $c > \frac12$ will do for large enough $x$.

I don't know why the textbook talks about "for all $x > 2$" when all you need is "for all large enough $x$."