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I was solving some questions from an analysis book and thinking about this question:

If we should give an example of real sequences with partial limits set $\{0,1\}$ and $[0,1]$ what whould be the sequences ?

I am not sure if $0,1,0,1,\ldots$ sequence satisfies it? Because Inf of the sequence is $0$ for $\{1,0,1,0,1,\ldots\}$ and Sup is 1.

Any comments ?

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Sequence $0,\,1,\,2,\,0,\,1,\,2, \, \ldots$ has partial limits set $\{0,\,1,\,2 \}$ and its supremum is $2$ –  M. Strochyk Oct 4 '12 at 18:57
@M.Strochyk I thought that for the partial limits we have to take a sub-sequence and I took 0,1,0,1,0,.... Then for this question it would be right if I take the sequence 0,1,0,1,0,... –  DreamLighter Oct 4 '12 at 19:02
Does "partial limits set" mean "set of limit points"? –  Trevor Wilson Oct 4 '12 at 20:43
When you say "satisfies it", do you mean "satisisfies the condition that its set of limit points is $\{0,1\}$", or "satisisfies the condition that its set of limit points is $[0,1]$"? –  Trevor Wilson Oct 4 '12 at 20:45
Do you know that $\mathbb Q$ is countable? –  Henning Makholm Oct 4 '12 at 21:08
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