# The relationship between indirect utility and expenditure functions

I am trying to understand the fact that $$e(p, v(p,y)) = y$$. There is a proof in the text Advanced Microeconomic Theory (Jehle and Reny) that states the following:

Because $$u(·)$$ is strictly 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, $$U = [u(0), u^b)]$$ for $$u^b > u(0)$$, and where $$u^b$$ may be either finite or $$+∞$$.

To prove, fix $$(p, y) ∈ R^{++}_n × R^+$$. We know $$e(p, v(p, y)) ≤ y$$. We would like to show in fact that equality must hold. So suppose not, that is, suppose $$e(p, u), where $$u = v(p, y)$$. Note that by definition of $$v(·), u ∈ U$$, so that $$u. By the continuity of $$e(·)$$ from, we may therefore choose $$ε > 0$$ small enough so that $$u + ε, and $$e(p, u + ε).

I don't understand the point below:

Note that by definition of $$v(·)$$, $$u ∈ U$$, so that $$u.

$$v(·)$$ denotes the maximum 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?

Even if $$u^b$$ is finite, it can never be achieved. This is what is meant by "does not attain a maximum". Rather, $$u(x)$$ approaches $$u^b$$ from below as $$x \to \infty$$. This is because $$u$$ is strictly increasing. If we had $$u(x_*) = u^b$$ for some $$x_*$$, then we would have $$u(x_* + 1) > u^b$$ and $$u^b$$ could not be a bound. This is why $$U$$ is written as the half-open interval $$[u(0), u^b)$$.