Every positive number has the unique square root I have found the following proof in Rosenlicht's Introduction to Analysis. I don't quite get why $(y-\epsilon)^2 < a < (y+\epsilon)^2$. Could someone explain it to me? Thanks. Any shorter proof or graphical intuition is also very welcome.


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*"There are numbers greater than $y-\epsilon$ in $S$" means that there exists $x\in S$ such that $y-\epsilon < x$ and so $(y-\epsilon)^2 < x^2 < a$, because $x \in S$.

*$y+\epsilon\notin S$ implies $(y+\epsilon)^2 > a$ by definition of $S$.
