Given the following sequence:

$\displaystyle x_0 = 1, x_1 = \sqrt{3+1}, x_2 = \sqrt{3+\sqrt{4}}, x_3 = \sqrt{3+\sqrt{5}},$

$\displaystyle x_4 = \sqrt{3+\sqrt{3+\sqrt{5}}}, x_5 = \sqrt{3+\sqrt{3+\sqrt{3+\sqrt{5}}}} \ldots$

Prove the above sequence converges and determine the limit.