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Let $\{a_n\}$ be a sequence of reals such that $a_n=1$ or $a_n=-1$.

Let $A=\{n\in \mathbb{N}|a_n=1\}$ and $B=\{n\in \mathbb{N}|a_n=-1\}$.

Suppose $A$ is equipotent with $B$. That is, $|A|=|B|=\aleph_0$.

Here, how do i prove that partial sum of $\sum a_n$ form a bounded sequence?

Or is it false?

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    Katlus: One should ask that $A$ has asymptotic density equal to 1/2 (and that so has $B$) but even this necessary condition is not sufficient.2012-08-29

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