Let $C([0,1])$ be equipped with the supremum norm $\|\cdot\|_{C([0,1])}$. For each $n \in \mathbb{N}$, let $L_n: C([0,1]) \to C([0,1])$ be a positive linear operator, meaning that $g \geq 0$ pointwise on $[0,1]$ implies $L_n g \geq 0$ pointwise on $[0,1]$.
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Define the test functions $e_0,e_1,e_2 \in C([0,1])$ by $e_0(t)=1$, $e_1(t)=t$, and $e_2(t)=t^2$. Suppose that