MATH.20:4.1 - Choose the quantity or object and its order

State what is unknown: a number, a function, a set, an error, an optimal value, or another mathematical object. State the comparison relation. Numerical order, pointwise order between functions and set inclusion support different conclusions.

For a numerical bound L≤q≤U, say which q is being bounded and under which assumptions. For an enclosure L⊆S⊆U, L consists of established members of S, while U contains every possible member under the stated description. Cardinality alone does not establish either inclusion.

For an optimum, distinguish the optimal value from an object attaining it. A bounded nonempty set of real values has an infimum and supremum, but an attaining object needs a further argument. For example, the interval (0,1) has infimum 0 and contains no minimum.

Select the needed direction and strength. To show q≤t, an upper bound at most t suffices. To refute that claim, a lower bound greater than t suffices. If the bounds straddle t, they leave this question open.