X Banach, Y

Use "if absolute convergence implies convergence in a normed space, then the space is a Banach space." If $\sum‖x_n+Y‖<∞$ Since $‖x_n+Y‖=\inf_y‖x_n+y‖$, $∃y_n$ s.t. $‖x_n+y_n‖≤2^{-n}+‖x_n+Y‖$ $\sum‖x_n+y_n‖≤1+\sum‖x_n+Y‖$