Q: Prove the proposition P(1) , where P(a) is the proposition “If n is a positive integer, then n² ≥ n.” What kind of proof did you use?
Ok! I'll write instead of
Consider the formula if is false or if is true, then is true.
is true, therefore the implication is true: we've proved
Another way to prove (and even more) is to say that the proposition is a tautology (i.e. is always true). Indeed:
If isn't a positive integer, then is false and the implication is true: is true.
If is a positive integer, because the multiplication on is compatible with the order. So the second part of the implication is true, and again is true.
In conclusion, whatever is (a number, a matrix, a topological space...), is true. In particular, is true.