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To solve this problem, let's break down the given conditions and then prove the statement step by step.

### Given:
1. \( f(y) \) is continuously differentiable on \((-\infty, +\infty)\).
2. \( f(y) < 0 \) for \( y \neq 0 \).

We need to prove that for the differential equation
\[ \frac{dy}{dx} = f(y), \]
the solution \( y = y(x) \) with the initial condition \( y(x_0) = y_0 \) exists on the interval \([x_0, +\infty)\) and that
\[ \lim_{x \to +\infty} y(x) = 0. \]

### Proof:
1. **Existence and Uniqueness**:
   Since \( f(y) \) is continuously differentiable, it satisfies the conditions of the Picard-Lindelöf theorem. Thus, for the initial value problem \( \frac{dy}{dx} = f(y) \) with \( y(x_0) = y_0 \), there exists a unique solution \( y = y(x) \) in some interval around \( x_0 \).

2. **Behavior of \( f(y) \)**:
   Given that \( f(y) < 0 \) for \( y \neq 0 \), \( y \) decreases when \( y \neq 0 \). Therefore, if \( y(x) \) starts at \( y_0 \neq 0 \), it will move towards 0 as \( x \) increases.

3. **Asymptotic Behavior**:
   To show that \( \lim_{x \to +\infty} y(x) = 0 \), consider the behavior of the solution as \( x \to +\infty \).

   Assume \( y \) does not approach 0. Then there exists some \( \epsilon > 0 \) and a sequence \( x_n \to +\infty \) such that \( |y(x_n)| \geq \epsilon \). However, since \( f(y) < 0 \) for \( y \neq 0 \), \( y \) should be decreasing and bounded away from 0, leading to a contradiction because \( y(x) \) must cross any positive threshold infinitely often. This contradiction implies that \( y(x) \) must approach 0 as \( x \to +\infty \).

4. **Detailed Analysis**:
   By integrating the differential equation, we get:
   \[ x - x_0 = \int_{y_0}^{y} \frac{dy}{f(y)}. \]

   Since \( f(y) < 0 \) for \( y \neq 0 \), the integral \( \int_{y_0}^{y} \frac{dy}{f(y)} \) is well-defined and finite as \( y \) approaches 0. Hence, \( x \to +\infty \) as \( y \to 0 \), implying that \( y(x) \) approaches 0 as \( x \to +\infty \).

Therefore, the solution \( y = y(x) \) with the initial condition \( y(x_0) = y_0 \) exists on the interval \([x_0, +\infty)\) and satisfies \( \lim_{x \to +\infty} y(x) = 0 \).

Would you like any more details or have any questions about this proof? Here are some questions you might consider:

1. Can you explain the Picard-Lindelöf theorem in more detail?
2. How does the condition \( f(y) < 0 \) influence the behavior of the solution?
3. Can you show the integration steps in more detail?
4. What are other methods to prove the asymptotic behavior of differential equations?
5. How would the solution change if \( f(y) > 0 \) for \( y \neq 0 \)?

**Tip:** Always ensure to verify the conditions for the existence and uniqueness theorems before applying them to differential equations.

Learn how to prove the existence and asymptotic behavior of a solution to a differential equation using the Picard-Lindelöf theorem. This problem explores advanced concepts in differential equations and requires understanding of existence and uniqueness theorems. Suitable for advanced undergraduate students.

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