The definition of least upper bound I am working with states that 

> 
> If $A\subset F$, where $F$ is an ordered field, then an element $a\in F$
>  is a **least upper bound** if 
> 
> 
> $1$. $a$ is an upper bound of $A$;
> 
> 
> $2$. $a\leq b$ for every upper bound $b$ of $A$.
> 
> 
> 

So say I wish to find the least upper bound of $A=\{a\in\mathbb{Q} \ | \ a<3\}$. Here $F=\mathbb{Q}$, and the way we define the least upper bound does not restrict it, if it exists, to being in $A$. I understand that. What I *don't* understand is how we know that in this case the least upper bound of $A$ is $3$. Does $A$ not have some "ending point" that is really close to $3$, but is not identical to $3$? Or is it because we could keep inching closer and closer to (but never actually reach) $3$ that a *least* upper bound is not found until we actually leave $A$ and choose $3$?

I guess I am having a hard time writing out my thoughts mathematically in this case why $3$ is the least upper bound. Perhaps someone can help me in this regard?