Suppose you have the following:
$A=\{\sin(\theta)|\theta \in [0,\pi]\}$
I need to find $\sup A$ and $\inf A$ and I need to answer if these are in $A$, but I'm having trouble finding out, how I should approach this.
I know that $\sup$ is the lowest number that is bigger or equal to all other numbers in $A$, but then since $\sin(\pi)=0$ then the highest number must be $\sin\left(\frac{\pi}{2}\right)=1$ and then the lowest number must be $\sin(0)=0$. So by that logic:
$\sup A = 1$
$\inf A = 0$
But my problem is, that I'm not certain that this is correct. So I'm posting here to find out if I have misunderstood something about the prospects of $\sup$ and $\inf$ or if I'm on the right track?
You are correct.
Note that $a\leq1$ for every $a\in A$, i.e. $1$ is an upper bound of $A$, and secondly that for every $\epsilon<1$ we have $\epsilon<1\in A$, so $A$ has no smaller upper bound. That means that $1$ is the least upper bound of $A$. Notation $\sup A=1$.
Note that $a\geq0$ for every $a\in A$, i.e. $0$ is a lower bound of $A$, and secondly that for every $\epsilon>0$ we can find some $a\in A$ with $a<\epsilon$, so A has no larger lower bound. That means that $0$ is the greatest lower bound of $A$. Notation $\inf A=0$.