Existence of Pareto Optima

Dividing by $||y^k_1||$ and taking a limsup of both sides, \begin{equation*} \sum_m\bar y_m\geq 0 \end{equation*} Assumption 3 then implies that $\sum_m\bar y_m=0$, so \begin{equation*} \bar y_1=-\sum_{m\geq 2}\bar y_m. \end{equation*} $||\bar y_1||=1$, so $y,\sum_{m\geq2}y_m\neq 0$. To conclude, $\bar y_1+0+\cdots 0=\bar y_1\in Y$ and $0+\sum_{m\geq 2}\bar y_m\in Y$, so $-\bar y_1\in Y$, contradicting 3. ∎