Paul
Reputation
Next privilege 50 Rep.
Comment everywhere
 Jun 12 comment Limits of Subsequences Is there any exceptions where the result be not true? Jun 12 comment Limits of Subsequences So the method of proof works also in the case if $s_{n}=t_{n},\forall n$, right! Jun 12 awarded Scholar Jun 12 awarded Commentator Jun 12 accepted Limits of Subsequences Jun 12 comment Limits of Subsequences Oh ok, that's fine now, Thank you! Jun 12 comment Limits of Subsequences Are you sure about the index of $s'_{n}$ in the last line above? Jun 12 asked Limits of Subsequences Jun 12 comment Sequence space and limits Is it true the other way; given a sequence $s$, there is a sequence $s'$ such that $s\subset s'$ and $\lim s/s'=0$? Jun 12 comment Sequence space and limits So is there any case where such sequence could exist? For example, any conditions on $S$ ? Jun 12 comment Sequence space and limits But I think at some point in the process of taking subsequences all the subsequences will behaves in a same rate, is it true? Jun 12 comment Sequence space and limits Is it possible to multiply the ratio $\frac{s'}{s}$ by a fixed sequence $x_{n}$ converging to zero, not necessary from $S$, to make the problem true? Jun 12 asked Sequence space and limits Jun 12 revised Set of sequences added 8 characters in body Jun 12 revised Set of sequences added 79 characters in body Jun 12 comment Set of sequences So, any help!!? Jun 12 comment Set of sequences It just came to my mind: if there is a sequence $\{s_{n}\}\in S$ which converges to 0 then there will be many such sequences in $S$ by taking subsequences of that $\{s_{n}\}$, which all will convereg to 0. Jun 12 revised Set of sequences deleted 1 characters in body Jun 12 revised Set of sequences added 100 characters in body Jun 12 comment Set of sequences @Arturo: $\{h_{n}\}$ is any sequence of functions in $H$. So to make the definition clear, $S=\{\,\{s_{n}\}: s_{n}=\sup_{\mathbb R}|h_{n}(x)|, \{h_{n}\}\subset H, s_{n}\to 0\}$.