Formal logic clarification How would one define all $x,k,r$ in the following? Do I need to define that both $x,k \in \mathbb{R}$ or does just defining that $k \in \mathbb{R}$ naturally imply that $x \in \mathbb{R}$ (or vice versa) simply using the comparison $x > k$?
So far I have come up with this:
$$ (\forall l \in \mathbb{R})(\exists k \in \mathbb{R})[(x > k \wedge x \in \mathbb{R})f(x) < l] $$
Really not sure how I'd write this with so many variables involved! Any thoughts are appreciated, thanks!
 A: If I'm using the format $(\text{bounded quantification})\text{predicate}$, then $$(\forall r\in\Bbb R)(\exists k\in\Bbb R)(\forall x>k) f(x)<r$$
Adding square brackets around the last quantified statement is irrelevant. The last three you've written are just wrong.
A: The first one should be understood as you intend to mean.
$\forall x \in \mathbb{R}$  is a shorthand for $\forall x (x\in \mathbb{R} \to \ldots)$ and $\exists x \in \mathbb{R}$ for $\exists x (x \in\mathbb{R}\land \ldots )$. So without any of these shorthands, the first one will become

*

*$\forall r (r\in\mathbb{R} \to \exists k (k\in\mathbb{R}\land \forall x (x\gt k \to f(x) < r)))$
The 2nd and 3rd are wrong in $x$ quantified twice, the 4th in $x$ not being quantified.
A: $$\text{ as } x \text{ tends to } \infty, \;f(x) \text{ tends to }-\infty $$ can be translated as $$\forall r<0 \;\:\exists k>0 \;\:\forall x>k \quad f(x)<r.$$
Unabbreviated: $$\forall r[\,r<0\implies\exists k(\,k>0\,\land\,\forall x[x>k\implies f(x)<r]\,)\,].$$
