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Let $\{a_n\}_{n \in \mathbb{N}} \in \mathbb{R_+}$ a sequence of real positive numbers such that $$ \lim_{n \to \infty} a_n = a $$ and $0<a<\infty$

I would like to know if is true that $\exists m \in \mathbb{N} : a_n \leq a_m \forall n \in \mathbb{N} $

Thanks

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No: let $a_n=1-\frac{1}{n}$. Then $\lim_{n\to\infty}a_n=1$, but the set $\{a_n:n\in\mathbb{N}\}$ has no maximum.

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