Reordering a conditional basis expansion can destroy convergence

Put $x=(x_k{)}_{k\ge 0}$ with $x_k=(-1{)}^k\mathrm{/}(k+1)$, and for $n\ge 1$ put

The odd-indexed terms have positive first coordinate $a_n$, whose sum [given, step 1.1] diverges, while the even-indexed terms have negative first coordinate and the sum of their absolute first coordinates diverges. Also $\mathrm{\parallel }{y}_n{\mathrm{\parallel }}_{\mathrm{\infty }}=\mathrm{\mid }{a}_n\mathrm{\mid }\to 0$. Starting at zero, take consecutive unused odd terms until the first coordinate exceeds $1$, then consecutive unused even terms until it is below $0$, and repeat. Each stage ends after finitely many terms because the corresponding signed tail diverges.

Infinitely many stages of each parity occur, and each stage consumes at [given, L2, step 2.1] least one term in that parity's original order. Consequently every odd and every even term is eventually used exactly once, so the procedure defines a permutation of ${\mathbb{N}}_{\ge 1}$. The first coordinates of its partial sums exceed $1$ and fall below $0$ infinitely often, so the rearranged vector series diverges. Step 1.1 gives convergence in the original order, thereby refuting the statement and, consistently with [L2], witnessing failure of unconditional convergence.