How can I prove every positive real number has square root? Does it ask that we can express any positive real number as square root of something? like 4 is equal to square root of 16?

Hint: Consider the set $\{x\in \Bbb{R}_{\geq 0} \colon x^2<y\}$. By the completeness axiom, there is a least upper bound of this set, call it $x_0$. 
If $x_0^2<y$, then by density of rationals, there is some rational number $q$ in $(x_0^2,y)$. Does this make sense? 
If $x_0^2>y$. What happens?
So there's only one choice left.
