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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 ...

Clash Royale CLAN TAG #URR8PPP up vote 2 down vote favorite 2 Is it possible to list all partial recursive functions such that every function appears only once in the enumeration. More precisely, does there exist (total) recursive function $f:mathbbN rightarrow mathbbN$ such that for any two different values $a,b in mathbbN$ (where $ane b$) the following two properties are satisfied: (1) $phi_f(a) ne phi_f(b)$ (the two functions on the left and right hand side aren't the same) (2) For any possible partial recursive function $g:mathbbN rightarrow mathbbN$ there exists some value $N in mathbbN$ such that $g=phi_f(N)$ (the two functions on the left and right hand side are the same) $phi_x$ denotes the function computed by the program corresponding to index $x$ (under a reasonable 1-1 correspondence between the collection of programs and $mathbbN$). computability share | cite | improve this question edited Aug 30 at 11:12 asked Aug 30 at 9:34 ...