Question regarding an $l^\infty$ space

Question

When we consider some $x=${$x_n$} in the $l^\infty$ space, we know that we must have, for each x, sup $|x_n|$ < $\infty$. This implies that x is bounded. However, does this mean that there must exist some constant $C$ such that $|x_n|$ $\leq$ $C$ $\forall$ $x\in l^\infty$, $n\in \mathbb{N}$?

Answer 1

The answer is no. If such a constant $C$ would exist, then the constant sequence $x_n=C+1$ would not belong to $l^\infty$.