# How do I formally prove a universal implication?

A textbook I am reading (Discrete Mathematics and its Applications) went from introducing formal propositional and predicate logic (including popular rules of inference like Modus Ponens and Universal Generalization) to introducing direct methods of proof for theorems of the form ∀n(P(n)->Q(n)).

Apparently, most mathematical proofs of any kind of theorem are “informal” and omit many logical rules of inference and argumentative steps for the sake of conciseness. However, because the textbook doesn’t provide even one example of a detailed “tedious” proof that expresses most or all rules of inference and axioms used in the proof, though I have a general idea of the connection between the two, I have been struggling to fully tie together the ideas of formal logic to the ideas of mathematically proving theorems of the form ∀n(P(n)->Q(n)). Can anyone provide an example of a detailed mathematical proof of a simple theorem that omits few (if any) logical steps in the argument? I have personally struggled with (as a personal exercise) meticulously proving the theorem “for all integers, if n is odd then the square of n is odd”, but any logically detailed argument proving a simple theorem similar to that would be very useful.

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