real analysis - How I can prove that the sequence $sqrt{2} , sqrt{2sqrt{2}}, sqrt{2sqrt{2sqrt{2}}}$ converges to 2?

Prove that the sequence $\sqrt{2} , \sqrt{2\sqrt{2}}, \sqrt{2\sqrt{2\sqrt{2}}} \ $ converges to $2$.

My attempt

I proved that the sequence is increasing and bounded by $2$, can anyone help me show that the sequence converges to $2$?
Thanks for your help.

Answer

Another prove:

Notice that:
$a_1 = 2^{1/2},\ a_2 = 2^{3/4},\ a_3 = 2^{7/8}$, and so on, thus,

$$a_n = 2^{(2^n-1)/2^n} = 2^{1-1/2^n}$$

Taking limit as $n$ tends to infinity, we have that

$$\lim_{n \rightarrow \infty} a_n = 2^1 = 2$$