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Question: To show that a banach space $E$ contains an isometric copy of $\ell^{\infty}(2),$ is it enough to show that there exists $e_1,e_2 \in E$ with $\| e_1\| = \|e_2\|=1$ such that for any $\alpha,\beta \in \mathbb{R},$ $\|\alpha e_1 + \beta e_2 \| \leq \max\{ |\alpha|,|\beta|\}?$
No, it means that $E $ has elements $e_1$ and $e_2$ such that $$ \|\alpha e_1+\beta e_2\|=\max\{|\alpha|,|\beta|\}$$ for all coefficients $\alpha $ and $\beta $.