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If a have a sequence of real numbers ordered from greatest to smallest $Y_o\geq Y_1 \geq...\geq Y_k$ and I divide each number of the sequence by $\delta<0$, is the order of the sequence flipped, i.e. $\frac{Y_0}{\delta}\leq \frac{Y_1}{\delta}\leq...\leq \frac{Y_k}{\delta}$?

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Yes, multiplying or dividing by negative numbers reverses inequalities. On the other hand, multiplying or dividing by positive numbers preserves inequalities.

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