show $\sup(\sin(x))=1$ and $\inf(\sin(x))=0$

Question

Can someone help with this:

I have:

$$A = \{ \sin(x)\mid x ∈ {]}0, \pi{[} \,\} $$

I have to find $\inf A$ and $\sup A$.

Answer 1

we have

$$\forall x\in]0,\pi [\;\;0 <\sin(x)\leq1$$

but $$\sin(\frac {\pi}{2})=1\implies \sup A=1$$ on the other hand $$\lim_{n\to+\infty}\sin (\frac {1}{n})=0 \;\;\implies \inf A=0.$$