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Utility, or usefulness, is the (perceived) ability of something to satisfy needs or wants.

The following is a proof that the indirect utility function is nonincreasing in prices, but I can't understand the last step. How do they conclude that $v(p_1, y) \ge$ from the previous reasoning … ? Consider $p_0\ge p_1$ and let $x_0$ solve the utility maximisation problem when $p = p_0$. Because $x_0\ge 0$, $(p_0 − p_1) · x_0 ≥ 0$. Hence, $p_1·x_0 ≤ p_0·x_0 ≤ y$, so that $x_0$ is feasible for the utility maximisation problem when $p = p_1$. We conclude that $v(p_1, y) ≥ u(x_0) = v(p_0, y)$. …
increasing on $R_n^+$, it attains a minimum at $x = 0$, but does not attain a maximum. Moreover, because $u(·)$ is continuous, the set $U$ of attainable utility numbers must be an interval. Consequently … utility that can be achieved for given prices and wealth, and I do not see why we cannot have $u = u^b$ such that $u ≤ u^b$ since $u^b$ may be finite. What am I missing here? …