The problem given in the image is in Chinese and translates to:

"13. Let the sequence $\{x_n\}$ be defined by $x_1 = 1$ and $x_{n+1} = \sqrt{2x_n}$ for $n = 1, 2, \ldots$. Prove that $\{x_n\}$ converges, and find $\lim_{n \to \infty} x_n$."

Solution:

Let's start by investigating the convergence of the sequence $\{x_n\}$.

1. Initial Condition and Recurrence Relation: $x_1 = 1, \quad x_{n+1} = \sqrt{2x_n}$

2. Monotonicity: We need to show that the sequence is either increasing or decreasing.

   • Suppose $x_n \leq x_{n+1}$: $x_n \leq \sqrt{2x_n}$ Squaring both sides: $x_n^2 \leq 2x_n \implies x_n^2 - 2x_n \leq 0 \implies x_n(x_n - 2) \leq 0$ Since $x_1 = 1$, and assuming $x_n \leq 2$, we have $x_n \leq 2$.

   • Suppose $x_n \geq x_{n+1}$: $x_n \geq \sqrt{2x_n}$ Squaring both sides: $x_n^2 \geq 2x_n \implies x_n(x_n - 2) \geq 0$ Since $x_1 = 1$, and assuming $x_n \leq 2$, the sequence does not initially decrease.

   This suggests $x_n$ lies between 1 and 2 and is decreasing after the first term.

3. Boundedness: The sequence is bounded below by 0 and above by 2.

4. Convergence: Since $\{x_n\}$ is bounded and monotonic, by the Monotone Convergence Theorem, it must converge.

5. Finding the Limit: Let $\lim_{n \to \infty} x_n = L$. Then: $L = \sqrt{2L}$ Squaring both sides: $L^2 = 2L \implies L^2 - 2L = 0 \implies L(L - 2) = 0$ Thus, $L = 0$ or $L = 2$.

   Since $x_n > 0$ for all $n$, $L = 0$ is not possible. Therefore, the limit is: $\lim_{n \to \infty} x_n = 2$

Conclusion:

The sequence $\{x_n\}$ converges, and its limit is: $\lim_{n \to \infty} x_n = 2$

Do you have any questions or would you like further details on any part of the solution? Here are some additional questions you might consider:

1. How do you determine if a sequence is bounded?
2. What is the Monotone Convergence Theorem?
3. How can you prove that a sequence is monotonic?
4. Can a sequence have more than one limit point?
5. What are some examples of sequences that do not converge?

Tip: When proving convergence of a sequence, always check for boundedness and monotonicity. These properties are crucial for applying the Monotone Convergence Theorem.