If $(a_n)$ diverges, then when does it tend to infinity
Clash Royale CLAN TAG #URR8PPP up vote 0 down vote favorite Let $(a_n)$ be a sequence of real numbers. Consider the two points, 1.$quad$ $(a_n)$ diverges, 2.$quad$ $(a_n)$ tends to plus infinity. Using the fact that an unbounded sequence diverges, the second point implies the first one. What conditions should $(a_n)$ have to make first point to imply the second one? It seems obvious to me that $(a_n)$ has to be non-negative eventually or increasing eventually, but I do not know how to prove it. Example : Let $(b_n)$ be a sequence of positive numbers, and define $s_N=sum_n=1^Nb_n$. If $sum_n=1^inftyb_n$ diverges, then it must be the case that $s_Ntoinfty$ as $Ntoinfty$, which is reasonable. But I want to know the proof. This is why I asked a question. real-analysis convergence share | cite | improve this question edited Aug 30 at 9:43 asked Aug 30 at 9:33 UnknownW 917 8 21 3 I don't think there's really a way to ...