This is the economic paper: Hopkins and Kornienko (2004): Running to Keep in the same place. The picture shows the proof of the proposition 1 in this paper. In this proof, I don't understand how the mean value theorem is applied.
Here is my attempt to understand the proof and I am considering the first inequality: By mean value theorem, $$V(x(z), z-px(z))(\alpha + G(z)) - V(x(\check{z}), z - px(\check{z}))(\alpha + G(\check{z})) - [(x(z) - x(\hat{z})][V_1(x(\tilde{z}), z - px(\tilde{z})) - pV_2(x(\tilde{z}), z - px(\tilde{z}))](\alpha + G(z))=0, $$
where $\tilde{z} \in (z, \hat{z})$. Then, I think they try to make this bigger or equal than 0:
$$ V(x(z), z-px(z))(\alpha + G(z)) - V(x(\check{z}), z - px(\check{z}))(\alpha + G(\check{z})) \le V(x(z), z-px(z))[G(z) - G(\hat{z})], $$
and
$$-[(x(z) - x(\hat{z})][V_1(x(\tilde{z}), z - px(\tilde{z})) - pV_2(x(\tilde{z}), z - px(\tilde{z}))](\alpha + G(z)) \le -[(x(z) - x(\hat{z})][V_1(x(\tilde{z}), z - px(\tilde{z})) - pV_2(x(\tilde{z}), z - px(\tilde{z}))](\alpha + G(\hat{z})).$$
But, both are wrong because by assumption, $ V(x(z), z-px(z)) \ge V(x(\check{z}), z - px(\check{z}))$, and $G(z)$ is strictly increasing.
If this is wrong, I don't understand what's going on here. Am I missing something in MVT?
I appreciate if you give any help.