Hi I am teaching myself analysis and bought "Analysis - With an introduction to Proof" by Steven R. Lay. Now one of the practice problems is "Determine the truth value of each statement, assuming x, y and z are real numbers" $$ \forall x, \exists y \, and \, \exists z \, such\, that\, z>y\, implies\, that\, z>x+y $$
Now the solution manual says "True. Take z<=y. This makes "z>y" false so that the implication is true." Now I find this "solution" troubling because I thought the whole idea behind proving an implication was to assume the antecedent IS true. Is this a mistake by the author?