Predicate logic proof problem Where the domain of the variables are Real Numbers, determine the truth value for the following:
$$ \forall x \exists y(y^2-x<200) $$
I don't understand how to formally prove this problem. Since $y^2\geq 0$ it would stand to reason that any $x< 0$ could disprove this statement. I tried to prove it true by using one case where $x=3$ and $y=3 \therefore  9-3<900 $ which makes it True.
How should I tackle a problem like this?
Thanks

One way to understand the statement $$ \forall x \exists y(y^2-x<200) $$ is to think of it as a game.  One player takes the $\forall$ quantifiers and one takes the $\exists$ quantifiers.  The $\forall$ player is trying to make the statement at the end (the $y^2-x<200$) false, and the $\exists$ player is trying to make it true.  Each quantifier, $\exists$ or $\forall$, is one move in the game.
In this game, the $\forall$ player moves first, and picks a value for $x$.  Then the $\exists$ player moves second, and picks a value for $y$.  The entire statement, including the quantifiers, is true if the $\exists$ player can always make $y^2-x < 200$.
Suppose the $\forall$ player picks $x=17$.  Then the $\exists$ player can win by picking $y=10$, since $10^2 - 17 < 200$.
Suppose the $\forall$ player picks $x=2$.  Can the $\exists$ player pick a $y$ that still makes $y^2-x < 200$?
Can the $\forall$ player make a move that leaves the $\exists$ player without a winning reply?  That's the question you are supposed to answer.

Here's my favorite example:  The statement $$\forall x\exists y (\text{$y$ is the mother of $x$})$$ is true, because whoever $\forall$ picks in move 1, $\exists$ can win by picking that person's mother.  Suppose $\forall$ picks $x$ equal to Angela Merkel; then $\exists$ wins by picking $y$ equal to Angela Merkel's mother.
But the statement $$\exists y\forall x (\text{$y$ is the mother of $x$})$$ is false, because now $\exists$ must go first, and she has no winning move. Suppose she picks $y$ equal to Angela Merkel's mother.  Then $\forall$ can win by picking $x$ equal to someone who is not Angela Merkel or one of Merkel's siblings; say George W. Bush.  Of course if $\exists$ picks Barbara Bush in her move, $\forall$ must not pick George W. Bush; he should pick someone else, like say Sisqo.

You asked in comments for an example with an implication.  Let's consider $$\forall x\exists y (\text{$x$ is even}  \to \text {xy = 2}).$$  
The $\forall$ player moves first. If he picks an odd number for $x$ he loses immediately, because then regardless of what $\exists$ does, the implication $\text{$x$ is even}  \to xy=2$ is vacuously true (the antecedent is false, so the entire implication is true), and the $\forall$ player loses when the final statement is true.  So if $\forall$ wants to win he had better pick an even number!  Suppose he picks $x=2$.  Then $\exists$ can answer with $y=1$.  And suppose he picks $x=4$; then $\exists$ can answer with $\frac12$ (if that is allowed; it may be clear from context, or it may be stated explicitly what moves are allowed.)  But if $\forall$ picks $x=0$, he wins no matter what $\exists$ does, and so the quantified statement is false.
