Let $x(n)$ a sequence in real numbers. Consider $A=\sup \left\{ |x(n)| : n \in \mathbb{N} \right\} $ . Is $A$ a $\max$ at the same time? i.e. Is the equality $A=\max \left\{ |x(n)| : n \in \mathbb{N} \right\}$ correct ?
2026-04-06 16:51:18.1775494278
Is $\ell_\infty$ a $\max$?
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No.
Consider for instance $(x_n)_n$ with $$ x_n = 1-\frac{1}{n+1}\,. $$ The supremum of the sequence is $1$. Yet, the sequence has no maximum.