If $L$ and $M$ are non-empty sets and $l\le m$ for all $l\in L$ and $m\in M$, there exists a real number $\alpha$ such that $\alpha \ge l$ for all $l\in L$ and $\alpha \le m$ for all $m\in M$.

It then takes $L=\{x\in R:x^2<2\}$ and $M=\{x\in R:x^2>2\}$ in order to prove that $\sqrt{2}$ exists. My problem is mainly that I feel uneasy about the statement ' $\sqrt{2}$ exists' and was hoping for a convenient proof; instead, it appears to me that the completeness axiom has been invented in order to facilitate a proof. It was my understanding that axioms were meant to be blindingly obvious, so much so that they are impossible to challenge; however, it seems to me that the statement 'there exists a real number $\alpha$' in the completeness axiom is not obvious. For it is one thing to say that one can get as close to $\alpha$ as you like, it is another to say that $\alpha$ exists. With $\sqrt{2}$, for example, one can talk about a sequence of nested intervals which have $\sqrt{2}$ in common, yet $\sqrt{2}$ can never be found and in which case, how can we say it exists? Would anybody be able to shed some light on this? Or, is this just a problem with my understanding that I am going to have to live with!