My goal is to prove that some set $A \subset \mathbb{R}^+$ (or $A \in \mathbb{R}^+ \cup \{\infty\}$ if it is allowed to be written in this way) is nonempty.
Unfortunately, I cannot draw some element in $A$ because it is very difficult to know the nature of $A$.
Fortunately, suppose I know the infimum of $A$ cannot be infinite, that is, $$\inf A < c \text{ for some constant $c < \infty$. }$$
Is this equivalent to state that "$A \neq \emptyset$"? I know the infimum of the emptyset is $\infty$ by convention, but I am not sure this convention can be used to prove non-emptiness.
Any clues or reference would be very appreciated.
In addition
I am also aware of the theorem stating that "every nonempty set of real numbers that is bounded below has an infimum in $\mathbb{R}$". Is the converse of the theorem also true?