Paul

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seen Jun 13 '12 at 20:39

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
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12
revised Set of sequences
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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
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12
revised Set of sequences
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Jun
12
comment Set of sequences
@Arturo: $\{h_{n}\}$ is any sequence of functions in $H$. So to make the definition clear, $S=\{\,\{s_{n}\}: s_{n}=\sup_{\mathbb R}|h_{n}(x)|, \{h_{n}\}\subset H, s_{n}\to 0\}$.