Sequence in $L^{\infty}$

Question

How can I prove that if the sequence $(f_n)$ converges in $L^{\infty}(]0,1[)$ then the sequence $(||f_h||_{\infty})$ of real numbers is bounded?

Answer 1

Well, if $f_n$ converges in $L^\infty$, then there is a $f \in L^\infty$ such that $\|f_n-f\|_{L^\infty} \rightarrow 0$, i.e. $\|f_n-f\|_{L^\infty} \leq C$ (Standart Basic Analysis result).

Thus with triangle inequality:

$\|f_n\|_{L^\infty} = \|f_n-f+f\|_{L^\infty} \leq \|f_n-f\|_{L^\infty}+\|f\|_{L^\infty} \leq C + \|f\|_{L^\infty} \leq \tilde{C}$.