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The theorem in your image appears to involve the concept of limits of functions of two variables and their relation to iterated limits. Here's a breakdown of the theorem and the steps needed for a proof:

### Theorem 13.2.2 (Statement):
Let \( f(x, y) \) be defined in a neighborhood \( N_0((x_0, y_0), \delta_0) \) of the point \( (x_0, y_0) \) (where \( \delta_0 > 0 \)), and assume the following conditions are satisfied:
1. There exists \( A \) (which could be \( \infty \)) such that:
   \[
   \lim_{(x, y) \to (x_0, y_0)} f(x, y) = A;
   \]
2. For any fixed \( y \neq y_0 \) within the neighborhood \( U_0(y_0, \delta_0) \), the following limit exists:
   \[
   \lim_{x \to x_0} f(x, y) = \varphi(y),
   \]
   where \( \varphi(y) \) is a function of \( y \).

Then, it follows that:
\[
\lim_{y \to y_0} \varphi(y) = \lim_{y \to y_0} \lim_{x \to x_0} f(x, y) = A.
\]

### Strategy for the Proof:
The key idea in this proof is to use the fact that the double limit (the limit as \( (x, y) \to (x_0, y_0) \)) exists and equals \( A \), and to show that this implies the iterated limits also converge to \( A \).

#### Step-by-Step Proof Outline:

1. **Existence of \( \lim_{(x, y) \to (x_0, y_0)} f(x, y) = A \):**
   - By assumption, the function \( f(x, y) \) approaches \( A \) as \( (x, y) \to (x_0, y_0) \). This means that for every \( \epsilon > 0 \), there exists a neighborhood \( N_0((x_0, y_0), \delta_0) \) such that:
     \[
     |f(x, y) - A| < \epsilon \quad \text{for all} \quad (x, y) \in N_0((x_0, y_0), \delta_0).
     \]

2. **Limit for fixed \( y \neq y_0 \):**
   - For each fixed \( y \neq y_0 \), the limit \( \lim_{x \to x_0} f(x, y) = \varphi(y) \) exists. This means:
     \[
     \forall \epsilon > 0, \exists \delta(y) > 0, \text{ such that } |f(x, y) - \varphi(y)| < \epsilon \quad \text{for} \quad |x - x_0| < \delta(y).
     \]

3. **Relationship between the iterated limits and the double limit:**
   - Since the double limit exists, it implies that:
     \[
     \lim_{y \to y_0} \varphi(y) = \lim_{y \to y_0} \lim_{x \to x_0} f(x, y) = A.
     \]
     - For this, we use the fact that as \( (x, y) \to (x_0, y_0) \), the function \( f(x, y) \) approaches \( A \) uniformly in both variables.

4. **Conclusion:**
   - Since the double limit and the iterated limits both tend to the same value \( A \), we conclude that:
     \[
     \lim_{y \to y_0} \varphi(y) = \lim_{y \to y_0} \lim_{x \to x_0} f(x, y) = A.
     \]

### Summary of Proof:
The proof leverages the assumption that the limit of \( f(x, y) \) as \( (x, y) \to (x_0, y_0) \) exists and equals \( A \), and shows that the iterated limits (first in \( x \), then in \( y \)) must converge to the same value \( A \). The neighborhood structure and uniform convergence are essential for handling the two-variable limit.

---

Would you like further details or have any questions? Here are 5 related questions to explore:

1. What are the conditions for the existence of a double limit in two variables?
2. How does uniform convergence affect the outcome of iterated limits?
3. Can you provide an example where the double limit exists but the iterated limits do not match?
4. What are the implications of the result for continuous functions?
5. How does this result relate to Fubini's theorem for iterated integrals?

**Tip:** When evaluating limits in multivariable calculus, always check if the limit depends on the path to the point of interest.

如何证明定理13.2.2涉及的双变量函数极限与迭代极限的关系？

This page explains the proof of Theorem 13.2.2, which explores the relationship between the double limit of a function of two variables and iterated limits. It covers key concepts in multivariable calculus, including limits of functions and the conditions under which iterated limits match the double limit. Suitable for university-level students studying advanced calculus or real analysis.

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