Definition of Real Numbers in Infinity?

Question

Real numbers is a set of all subsets A $⊂$ $Q$ with this features:

1. A $ ≠ ∅$ , A $≠ Q$

2. A is closed from underneath, ( $∀$ x,y $∈ Q$) ( x < y $∧$ y $∈$ A) $⇒$ x $∈$ A

3. A doesn’t have the biggest element.

My question is how can A be closed from underneath in feature 2.? We can take smaller and smaller x and that will go to $-∞$, which is not closed.

Answer 1

"Closed" in this context does not mean that sequences have limits in the set. It means closed under the operation of "less than". You say just that in item (2).