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

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?