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?