We are given the recursive sequence defined by:

$a_{n+1} = \sqrt{2 + a_n}, \quad a_1 = \sqrt{2}.$

We are tasked with proving that the sequence is decreasing.

Step 1: Checking if the sequence is bounded

We will first check whether the sequence has an upper bound. Notice that:

$a_1 = \sqrt{2}.$

Since $a_{n+1} = \sqrt{2 + a_n}$, clearly $a_{n+1} \geq \sqrt{2}$ for all $n$, meaning the sequence cannot go below $\sqrt{2}$. Hence, the sequence is bounded below by $\sqrt{2}$.

Step 2: Prove that the sequence is decreasing

Now, to prove that the sequence is decreasing, we need to show that:

$a_{n+1} \leq a_n \quad \text{for all } n.$

This implies showing that:

$\sqrt{2 + a_n} \leq a_n \quad \text{for all } n.$

Step 2.1: Squaring both sides

To eliminate the square root, we can square both sides:

$(\sqrt{2 + a_n})^2 \leq a_n^2.$

This simplifies to:

$2 + a_n \leq a_n^2.$

Rearranging terms gives:

$a_n^2 - a_n - 2 \geq 0.$

Factoring the quadratic expression:

$(a_n - 2)(a_n + 1) \geq 0.$

Step 2.2: Analyzing the inequality

For $(a_n - 2)(a_n + 1) \geq 0$, there are two possibilities:

• $a_n \geq 2$, or
• $a_n \leq -1$.

Since we know from the recursive relation that $a_n \geq \sqrt{2} \approx 1.414$, the condition $a_n \geq 2$ never holds for our sequence, which starts below 2. Thus, for the inequality to hold, the sequence cannot be decreasing for values less than 2.

Conclusion:

The sequence is not decreasing since the inequality $a_{n+1} \leq a_n$ is not satisfied in the range of values $1.414 \leq a_n \leq 2$.

Would you like to review any part of this reasoning? Here are five related questions to deepen your understanding:

1. How do you rigorously show that a sequence is bounded?
2. Can a sequence converge if it is not monotonic?
3. What additional conditions are needed to prove that a sequence converges?
4. How does the initial value of $a_1$ affect the behavior of the sequence?
5. Can you use an alternate approach, like induction, to test if this sequence is decreasing?

Tip: When working with sequences, always check if they are bounded and monotonic to infer convergence.