Prove there exists some $\beta < 0$ for which $W(x) \leq \beta +\sin(x)$ on $[1,6].$

Question

Suppose that $W : [1,6] \rightarrow \mathbb{R}$ is continuous and $W(x) < \sin(x)$ on $[1,6].$ Prove there exists some $\beta < 0$ for which $W(x) \leq \beta + \sin(x)$ on $[1,6].$

I was thinking of defining a new function, $h(x)= W(x)-\sin (x)$, so $h(x)$ will also be continuous on $[1,6]$ and $h(x)<0$,now how to proceed?

Answer 1

The function $W-\sin$ is continuous and strictly negative; moreover $[1,6]$ is a compact set, thus by Weierstass it admits a maxmum $\beta<0$, thus $W\le\beta$, that is $W(x)\le\beta+\sin x$ as wanted.

Answer 2

So here is a detailed answer to the problem.