TITLE: If A entails B, and B entails C, does A entail C?
QUESTION [5 upvotes]: Or does C entail A?
This is a most simple question that bothers me why I don't understand the answer. Propositional logic is very difficult for me. Could you please help me out?

REPLY [4 votes]: From the comments, OP confirms that $X$ entail $Y$ means $X \implies Y$.
Then the OP's question is whether
$$\bigg((A \implies B) \land (B \implies C)\bigg) \implies (A \implies C)$$
holds.
Hint: See Hypothetical syllogism.