Give a proof outline of the specified method for each statement. Use only the propositional structure of the claim (that is, do not use the definitions of the technical terms in each statement). 1. If G is a finite group, then every subgroup of G is finite. (direct proof) 2. Let x, y ∈ R. If y ≤ x + µ for any positive real number µ, then y ≤ x. (proof by contraposition) 3. Let R be an integral domain. The only divisors of an irreducible element of R are its associates and the units of R. (direct proof)

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