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Optimal transport: the existence of an optimal pair of $c$-conjugate functions

I was also struggling with Exercise 2.36... I think that I am now able to solve it, although it seems that it is more difficult than it appears... The key seems to be the following theorem. Theorem: ...
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Optimal transport: how $\varphi^c$ can be written as $\varphi^c = \lim _{\ell \rightarrow \infty} \psi_{\ell}$?

Here is an example which shows that also the continuity of $\varphi$ does not help. Let $X = \mathbb R$ and $Y = \{0\}$. In the sequel, we will just drop the $Y$-argument. Let $c \equiv 1$, $\varphi \...
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