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Let $c$ be a limit point of a sequence of real numbers $\langle a_n \rangle$ and $d$ a limit point of $\langle b_n \rangle$. Is $c+d$ necessarily a limit point of $\langle a_n + b_n \rangle$?

My Question:

When considering this question, do I have to sum over the same index or can the indices for the different subsequences differ? My intuition is that the subsequences must be summed over identical indices, in which case I believe that the following example serves as a counterexample:

Let $\langle a_n \rangle = (-1)^n, \ \langle b_n \rangle = (-1)^{n+1}$. Then summing over even and odd indices, I get $0 \ne 2$ or $-2$, which is the sum of their limit points. Have I done this correctly?

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    Your argument is correct; but generally, the indices could differ. Take $a_n=b_n=(-1)^n$. $(a_n)$ has $1$ as a limit point and $(b_n)$ has $-1$ as a limit point but $(a_n+b_n)$ does not have $1+(-1)=0$ as a limit point.2012-08-14

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Your example is fine. Both $\langle(-1)^n:n\in\Bbb N\rangle$ and $\langle(-1)^{n+1}:n\in\Bbb N\rangle$ have $1$ as a limit point, but $\langle(-1)^n+(-1)^{n+1}:n\in\Bbb N\rangle=\langle 0:n\in\Bbb N\rangle$ converges to $0$ and so does not have $1+1=2$ as a limit point.

It doesn’t matter what subsequence of $\langle a_n:n\in\Bbb N\rangle$ has $c$ as limit or what subsequence of $\langle b_n:n\in\Bbb N\rangle$ has $d$ as a limit; all that matters is whether some subsequence of $\langle a_n+b_n:n\in\Bbb N\rangle$ has $c+d$ as a limit. In your example that’s not the case, so yours is a genuine counterexample to the conjecture.

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    Ah ok. Thank you all for your help, I am self-studying introductory real analysis and slowly getting the hang of the material.2012-08-14