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I was doing an exercise in a book which asked the question "what can be said about a divergent sequence which diverges to neither $\infty$ or $-\infty$?" I came up with the answer that it is bounded. I was wondering two things:
1. Is there anything more you can say about it?
The above led me to think about:
2. Is there a concept of a sequence of numbers converging to a certain range of numbers? and would this even be useful? I'm thinking then that the $\epsilon, N$ defn. of convergence to a limit would be a special case of the "convergence to a range of numbers" when the range has only one number in it.
Thanks for any thoughts/advice. :)


p.s. the exercise is not for a grade for a class, it's just out of curiosity.

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1 Answer 1

up vote 5 down vote accepted

Not necessarily bounded. Look at $1,0,2,0,3,0,4,0,5,0,\dots$.

And yes, if we have a sequence, we can look at its set of accumulation points. These are all numbers $x$ such that there is an infinite subsequence of our sequence that converges to $x$. Alternately, $x$ is called a limit point of the sequence. The concept is useful. One can alter the definition to throw in the possibility of $\infty$ and $-\infty$.

A sequence $(a_n)$ converges to a real number iff it has exactly one accumulation point.

Interesting example: Look at the sequence $\sin(1), \sin(2), \sin(3),\dots$. Then every number if $[-1,1]$ is an accumulation point of the sequence.

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