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Let $\{a_n\}$ be a real-values sequence which is convergent but not absolutely.

Let $P_n$ enumerate nonnegative terms and $Q_n$ enumerate negative terms.

It's clear that the set of $P_n$ and that of $Q_n$ are infinite.

Let $I(n) = n, \forall n\in \mathbb{N}$. Let $\{m_n\}$ and $\{k_n\}$ be subsequences of $I$.

Expand like follows;

$P_1, ... , P_{m_1} , Q_1 , ... , Q_{k_1} , P_{m_1 +1} , ...$

It's clear that this is a rearrangement of $\{a_n\}$, but how do i show this enumerates $\{a_n\}$ in meta language?

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1 Answer 1

All you need to do is say that since each element in your expanded sequence corresponds to exactly one element of the original series, you have that:

$$P_1, \dots , P_{m_1} , Q_1 , \dots , Q_{k_1} , P_{m_1 +1} , \dots=a_{\varphi(1)},a_{\varphi(2)},\dots$$

where $\varphi$ is a permutation of $\Bbb N$.

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