Is the supremum of a null sequence always a maximum?

Question

If the sequence $(z_n)$ is a complex null sequence, is it always true that $\sup_n|z_n| =|z_k|$, for some integer $k$? That is, the supremum is in fact a maximum of the sequence $(|z_n|)$?

Answer 1

Yes. If the sequence is constant, then obviously the maximum of $0$ is attained. Otherwise choose some $z_n \neq 0$, and apply the definition of convergence with $\varepsilon = |z_n| > 0$. Then, all but a finite number of points will be strictly less than $|z_n|$, leaving only finitely many that can be larger than $|z_n|$. Find the largest of this finite set, and that will be the maximum modulus of the sequence.

Answer 2

If $z_n=0$ for every $n$ then of course we have: $$\sup\{|z_n|\mid n\in\mathbb N\}=0=|z_1|$$

If $z_{m}\neq0$ for some $m$ then $n_0\geq m$ exists such that: $n>n_0\implies |z_n|<|z_m|$.

Consequently: $$\sup\{|z_n|\mid n\in\mathbb N\}=\max\{|z_1|,\dots,|z_{n_0}|\}\in\{|z_1|,\dots,|z_{n_0}|\}$$