Consider the town which can be reached from maximum number of other towns.$ (1)$ (If there are more than one such town we consider one of them) Call this town $ T$ Suppose it can be reached from $ k$ towns. Assume that there is a town which doesn't lead to $ T$.$ (2)$ Call this town $ X$. It is given that either $ T$ can be reached from $ X$ or $ X$ can be reached from $ T$. Because of assumption $ (2)$, the former can't be true. So the later must be true. So $ X$ can be reached from $ T$ But in that case $ X$ can be reached from all those $ k$ towns which lead to $ T$. So $ X$ can be reached from at least $ k+1$ towns. This contradicts the definition of $ T$ (in $ (1)$) So assumption $ (2)$ must be false. Hence there is no town which doesn't lead to $ T$. Proved.

Sorry for reviving an old thread, but I think there's a small mistake in the previous post. The question says: Quote: Prove that there is a town such that all remaining towns can be reached from it. The same approach works anyway.

For any towns $ x, y$ denote $ x \to y$ if town $ y$ can be reached from $ x$, and denote by $ C_x = \{y \ | \ x \to y\}$ the set of all towns that can be reached from $ x$ ($ C_x$ may even be empty). Clearly $ y \in C_x$ implies $ C_y \subseteq C_x$. With a finite number of towns, there exists (at least) a maximal set $ C_x$ with respect to the order relation $ \subseteq$ among sets. Assume $ y \not \in C_x$, so $ x \not \to y$, therefore $ y \to x$, so $ x \in C_y$, hence $ C_x \subseteq C_y$. By the maximality of $ C_x$ follows $ C_y = C_x$, whence $ x \to y$, contradiction. For an infinite number of towns this proof fails (even Zorn's lemma does not apply). A simple counterexample is given by towns $ x_z$ indexed by $ z \in \mathbb{Z}$, with one-way roads leading from $ x_z$ to $ x_{z+1}$, so $ x_s \to x_t$ if and only if $ s < t$. There is no town from which all others can be reached, neither one reached from all others.