calculus - Recursive square root problem

Give a precise meaning to evaluate the following:
$$\large{\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\dotsb}}}}}$$

Since I think it has a recursive structure (does it?), I reduce the equation to

$$p=\sqrt{1+p}$$
$$p^2=1+p$$
$$p^2-p-1=0$$
$$p=\frac{1\pm\sqrt{5}}{2}$$

Did I do this right?