Possible book correction or am I missing something? 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?

It is not a mistake; see the truth table here.
The main idea is that in the logical implication $p\Rightarrow q$, if $p$ is false, then the implication does not actually promise anything about $q$. My math prof at the time told the class: irrelevance does not imply falsehood.
In your problem, $z$ and $y$ are preceded by existential quantifiers, so we can choose $z$ and $y$, and thus we choose $z$ and $y$ such that $z\le y$, making the implication true.
