Moving limit into sup norm I need help understanding where I'm going wrong with this line of thought:
Assume $f_n$ converges pointwise to $f$, so $\lim \limits_{n \rightarrow \infty}f_n(x) = f(x) \forall x \in X$, then since the suprumum norm is a norm therefore continuous we can move a limit inside, like this:
$\lim \limits_{n \rightarrow \infty}\|f_n-f\|_\infty = \|\lim \limits_{n \rightarrow \infty}(f_n-f)\|_\infty = \|0\|_\infty = 0$
So all pointwise convergent sequences are also uniform convergent.
This is clearly not the case so where am I going wrong? 

The norm in this case is continuous in the sense that $u_{n}\rightarrow u$ in $L^{\infty}$, then $\|u_{n}\|_{L^{\infty}}\rightarrow\|u\|_{L^{\infty}}$, or we can write $\lim_{n}\|u_{n}\|_{L^{\infty}}=\|\lim_{n}u_{n}~(\text{limit taken in }L^{\infty})\|_{L^{\infty}}=\|u\|_{L^{\infty}}$.
