While thinking about Riemann's rearrangement theorem, the following question struck me :
Let $a_{n}$ be a converging real sequence, such as $\sum a_{n}$ is a conditionally converging series. For all $r \in \mathbb{R}$, does there exist $I \subset \mathbb{N}$ such as $\sum_{i \in I} a_{i} \rightarrow r$ ?
If so, is $I$ unique ?
Sorry if I'm missing something trivial, I'm a theoretical physics student, not a mathematician.
EDIT : corrected a mistake : $I$ is obviously a subset of $\mathbb{N}$