Monotone square-root recursion limit

Let $\{a_n\}_{n\ge1}$ be a sequence of real numbers such that $a_1 = \sqrt{6}$ and $a_{n+1} = \sqrt{6 + a_n}$ for $n \ge 1$. Consider the following statements: (I) $\{a_n\}_{n\ge1}$ is an increasing sequence. (II) $\lim_{n\to\infty} a_n = 2$. Which of the above statements is/are true?