There is a sequence $a_1, a_2, a_3 ...$ of real numbers $\{a_n\}\in\mathbb{R}$ that diverges to plus infinity. At least the first element is negative $a_1<0$. Is it possible to prove that only a finite number of numbers $\{a_n\}$ are negative? Is the condition $a_1\leq a_2\leq a_3\leq ...$ necessary for the proof?
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$\begingroup$$$\forall x\in \mathbb R \;\exists m_x\in \mathbb N \;\forall n>m_x\;(a_n>x)\implies$$ $$\implies \exists m_0\in \mathbb N\;\forall n>m_0\;(a_n>0)\implies$$ $$\implies \exists m_0\in \mathbb N\;\forall n\;(a_n\leq 0\implies n\leq m_0)\implies$$ $$\implies \exists m_0\in \mathbb N\;(\{n\in \mathbb N:a_n\leq 0\}\subset \{1,...,m_0\}).$$
$\endgroup$ $\begingroup$Yes , by definition, if the sequence converges to positive infinity, there is for every real number $R$ a time $n_0 \in \mathbb{N}$ so that $a_n \geq R$ for all later n and as you observe you don't have to use monotonicity of your sequence
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