So I need to find a function say $g(x)$ that upper bounds another function say $f(x)$; On that same note suppose $f(x)$ is a ceil function I am acquainted with the inequality that holds for ceil functions i.e:
$$x \leq \lceil x\rceil \lt x+1$$
considering the question as I am interested in the upper bound of the function $f(x)$ (here $\lceil x\rceil$) the right side of the inequality works for me $[\lceil x\rceil < x+1]$ but if I were to multiply a constant c on both sides right off the bat I can see $[c\times\lceil x\rceil \leq c\times(x+1)]$ when $c\geq 0$
Is the above inequality valid for ceil functions? (for any $c$ other than $0$)
Also, if the above is true then is :
$c\times \log \lceil x\rceil \leq c\times\log(x+1)$ , also true
P.S: I am pretty new to this site, apologies for not using correct notations.