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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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for finitely many limiting values $a_1,...,a_n$ use the sequence $a_1,...,a_n,a_1,...,a_n,a_1,...,a_n,...$ ie just repeat the values over and over. (i assume by partial limit you mean there exists a subsequence converging to that value) – yoyo Dec 15 '11 at 22:50
$a_1,a_1,a_2,a_1,a_2,a_3,a_1,a_2,a_3,a_4,\dots$ – Gerry Myerson Dec 15 '11 at 22:52
I don't really understand what you mean by that. Yes I'm referring to limits of sub-sequences. – nofe Dec 15 '11 at 22:53
yoyo answered the first. 1,2,3,1,2,3,1,2,3,1,2,3,... Can converge to 1, 2, or 3. Gerry gave an example for your second question. – The Chaz 2.0 Dec 15 '11 at 23:33
@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) – Nescio Dec 8 '12 at 9:55
up vote 1 down vote accepted

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.

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