Can someone explain this existence quantor defining real numbers?

Question

In school a teachers of mine gave us his slides and I came across a topic that we didn't really discuss a lot but is important for a test. In the linked image you can see the definition of real numbers but I couldn't really figure out what it means(the part with the existence quantor). It would be very nice if someone could explain what the part below and next to the quaontor means or just translate it into words.

(I'm not the best at English so excuse typos and by existence quantor I mean the big V if the english name is different)

definition of real numbers image

Answer 1

The symbol $\vee$ usually means the logical operation 'or', and anything below an operation symbol means a condition.

It means the set of all entities $x$ such that there is a sequence $(a_n)$ of rational numbers (i.e. $(a_n)\in\Bbb Q^{\Bbb N}$), such that $\lim_na_n=x$.

(at least one of the statements $x=\lim_{n\to\infty}a_n$ holds, when $(a_n)$ runs over $\Bbb Q^{\Bbb N}$.)

Note by the way, that $\pm\infty$ are not real numbers, though this definition might enable them (depending on how exactly $\lim$ was defined).

Answer 2

To me the definition seems to say: The set of all $x$ such that at least one of the statements "$x$ is the limit of a sequence of rational numbers $(a_n)_n$" is true.

The symbol is a logical disjunction, i.e. an "OR".

Answer 3

It means that every finite (bounded) limit of a sequence of rational numbers is defined to be a real number.

The key is that different sequences can have the same limit, and specifying the conditions for that to be the case.

The prize is that every bounded limit of a sequence of real numbers is then a real number. So you have a system complete under the operation of taking bounded limits.