Supreme and Infimus I need to know why this happens ... 
Reading a book just find that $$\sup_{y \in [0,2\pi]} \left \{ \inf_{x \in [0,2\pi]} \sin(x+y)  \right \}=-1$$

Regardless of the value chosen for $y$, $\sin(x+y)$ varies over its entire range of $[-1,1]$ if $x$ is allowed to range over a full period $[0,2\pi]$. Hence,
\begin{align*}
\inf_{x \in [0,2\pi]} \sin(x+y) = -1
\end{align*}
for every $y$. The supremum of the singleton $\{-1\}$ is of course $-1$. Hence
\begin{align*}
\sup_{y \in [0,2\pi]} \left\{ \inf_{x \in [0,2\pi]} \sin(x+y) \right\} = -1
\end{align*}
