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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.

Learn how to prove convergence and find the limit of the sequence $x_n = \sqrt{2x_{n-1}}$. This problem covers key mathematical concepts such as sequences, limits, convergence, and monotonicity. Suitable for advanced high school students.

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