$f_n[0,2 \pi] \to \mathbb{R}$ ,$f_n\overset{u}{\rightarrow} \sin $, I need to prove that $\sup _{x \in [0, 2 \pi]}f_n(x)$ and $\inf _{x \in [0, 2 \pi]}f_n(x)$ converges and find their limits.
I know that due to uniformly convergence, for $\epsilon=1$ there's $N$ that for every $n>N$ to all $x \in [0, 2\pi]$, $f_n(x)-f(x)|<1$, so $|f_n(x)|<2$ and bounded so $\sup_{[0,2\pi]}$ is finite. same case to the infimum. How should I go on in order to find their limits? ( 1 and 0?)
Thanks!