A singleton as domain sum of a series

Question

Consider the series $$e_1+\frac{1}{2}e_2-\frac{1}{2}e_2+\frac{1}{4}e_3-\frac{1}{4}e_3+\frac{1}{4}e_3-\frac{1}{4}e_3+\frac{1}{8}e_4-\frac{1}{8}e_4+\cdots-\frac{1}{8}e_4+\frac{1}{16}e_5-\cdots$$ in the space $\ell_2$. In example 2.2.1 of monograph Rearrangements of Series in Banach Spaces, V.M. Kadets & M.I.Kadets it is said that the domain of sums of this series consists of only the one point $e_1$, since the projection of this series on an arbitrary coordinate axis contains only finitely many nonzero terms, and a finite sum is not changed by rearrangements. Can someone explain more this reasoning to understand it? Thanks in advance.

Answer 1

Suppose some rearrangement of this series converged to a vector $v\neq e_1$, and choose an index $k$ such that these vectors differ in the $k$-th component. That is, $p_k(v)\neq p_k(e_1)$, where $p_k$ is the $k$-th projection. Then, since $p_k$ is linear and continuous, applying $p_k$ to every term of the rearranged series, we get a series of numbers converging to $p_k(v)$. But in fact, as you already mentioned in the question, the series we get in this way has only finitely many non-zero terms and converges to $p_k(e_1)$.