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Problem 73: (Limits Of Sequences And Limit Points Of Images)
Is a limit point of the image of a sequence equal to the limit of that sequence?
- Give an example of a sequence $(a_n)$ that converges to $L$, such that $L$ is a limit point of the image of the sequence.
- Give an example of a sequence $(a_n)$ that converges to $L$, such that $L$ is not a limit point of the image of the sequence.
- Make a conjecture about when limits of sequences and limit points of images of sequences are equal.
- Give an example of a sequence $(a_n)$ that does not converge, yet the image of the sequence has one (or more) limit points.
We'll have a class discussion on this one, rather than have someone present it. Please come with examples and a conjecture. If you choose to type this one up, you will need to prove any claims you make (about a sequence converging, or not, to what you say it does (which means you probably want to focus on typing up a different problem, as doing so could be quite time consuming if your examples are non trivial.
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