Recursive Sequence Monotonicity and Limit

Define a recursive sequence $a_1 = \sqrt{2}$ and $a_{n+1} = \sqrt{2 + a_n}$ for $n \geq 1$. Prove that the sequence is monotonic and bounded using the Monotone Sequence Theorem, and find its limit.

This problem is an excellent illustration of analyzing recursive sequences, which is a common topic in real analysis. To approach this problem, the Monotone Sequence Theorem will be a key tool. This theorem states that every bounded and monotonic sequence converges, which helps us determine the behavior at infinity of sequences defined recursively. The sequence in this problem begins with a specific value and is defined such that each term is determined by the previous one in a manner that builds upon itself in a predictable pattern.

The first step is to prove that the sequence is monotonic. To show this, you must verify whether the sequence is strictly increasing or decreasing by comparison of successive terms. In this context, using induction could be a suitable method to establish the necessary inequality for monotonicity. Next, demonstrating that the sequence is bounded requires identifying an upper or lower limit that the sequence cannot surpass. By applying these bounds and monotonic properties, the convergence of the sequence is assured by the Monotone Sequence Theorem.

Finally, to find the limit of the sequence as n approaches infinity, consider the behavior of the sequence's terms. Given its recursive nature, an equation might be derived by considering the scenario where both $a_n$ and $a_{n+1}$ approach the same limit. Solving this equation will yield the limit as a fixed point the sequence tends towards. This problem thus encapsulates key sequence behavior concepts such as recursion, monotonicity, boundedness, and convergence, making it a valuable component of real analysis studies.