Evaluating Continued Product of Radicals

Question

I am trying to find the following continued product

$y = \sqrt{p}.\sqrt{p+p.\sqrt{p}}. \sqrt{p+p.\sqrt{p+p\sqrt{p}}}...........\infty$

for $p=\frac12$

First I squared the LHS and RHS to get $y^2 = p.(p + p\sqrt p)(p + p\sqrt{p+p.\sqrt p})......$

Taking p common

$y^2 = p^n.(1 + \sqrt p)(1 + \sqrt{p+p.\sqrt p}).....$

As n tends to infinity $p^n$ should tend to zero for p=1/2. Hence y=0.

Is this the correct approach? Are any alternate methods possible?