Let the sequence $\left \{ x_{n} \right \}_{n\rightarrow 1}^{\infty }$ be defined by $x1=\sqrt{2}$ and $x_{n+1}=\left ( \sqrt{2} \right )^{x_{n}}$ for $n\geq 1$. Then which one of the following statements is true?
- The sequence $\left \{ x_{n} \right \}$ is monotonically increasing and $\underset{n\rightarrow \infty }{lim}\:x_{n}=2$
- The sequence $\left \{ x_{n} \right \}$ is neither monotonically increasing nor monotonically decreasing
- $\underset{n\rightarrow \infty }{lim}\:x_{n}$ does not exist
- $\underset{n\rightarrow \infty }{lim}\:x_{n}=\infty$