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I'm studying for a test in Calc I, and one of the practice problems is to "find a sequence with exactly 3 partial limits" and "find a sequence with an unlimited number of partial limits."

I have answers, they are both examples of such cases. I don't understand however how I can systematically create such a sequence.

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    @yoyo for a private case or the general case, how would you prove this is indeed the number of partial limits? (sorry for coming back to such an old question, but I got it as related when about to ask a similar question)2012-12-08

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The Chaz answered part of your question. Here's one with infinitely many partial limits: $ 1, \underbrace{1,2}, \underbrace{1,2,3}, \underbrace{1,2,3,4},\underbrace{1,2,3,4,5},\ldots . $

Here's another: $ \underbrace{\frac12}, \underbrace{\frac13,\frac23},\underbrace{\frac14,\frac24,\frac34},\underbrace{\frac15,\frac25,\frac35,\frac45},\ldots. $ Every number between $0$ and $1$ (inclusive) is a partial limit of this sequence.