How can I definitively determine whether a universal/existential statement is true or false? How can one look at a universal/existential statement and determine with absolute certainty whether it is true or false?  For example, is
$$∀x\in\mathbb R\,∀y\in \mathbb R\,∃z\in \mathbb R\,\left(z = \frac{x+y}2 \right)$$ true or false?
I'm pretty sure it's true, because any number divided by $2$ is going to yield some number (I tried some insane ranges just to check my sanity).  But how can I know for sure?  What about a more complex problem where a range of substitutions isn't a feasible test?
 A: As Adriano pointed out, this is simply because the real numbers are closed under addition and multipication (in this case, multiplication by $1\over2$). The reason you're struggling so much with a proof of this fact is because to really prove it you would need a definition of addition and multiplication, such as via Dedekind cuts or Cauchy sequences.
Intuitively, the proposition is of course obviously true. But yeah, the answer to question "how do I know for sure" is "by diving nearly as deeply into math as the subject allows". You're basically dealing with how we define exactly what a number is, and how to define addition and multiplication on that foundation. Look into construction of the real numbers, or construction of the rationals for a more accessible example.
A: This follows from the fact that the set of all real numbers is closed under addition and division by $2$. Since the order of the quantifiers is $\forall x \forall y \exists z$, you must choose any arbitrary $x,y \in \mathbb{R}$ and give an explicit $z\in \mathbb{R}$ that satisfies the condition (where $z$ is often a function of $x$ and $y$). Indeed, the given equation has already been solved for $z$, so this question was relatively easy.
