https://r2.erweima.ai/mathbot/image/1726333120849ujh9kouo.jpg

The image contains a series of exercises about the convergence of sequences. Let me help you understand and prove each statement one by one:

1. **First statement**:  
   *Demostrar: Si la sucesión \(a_n\) converge a \(b \in \mathbb{R}\), entonces la sucesión \(|a_n - b|\) converge a 0.*  
   
   **Proof**:  
   Since \(a_n \to b\), for every \(\epsilon > 0\), there exists \(N \in \mathbb{N}\) such that for all \(n > N\), \(|a_n - b| < \epsilon\).  
   This implies that the sequence \(|a_n - b|\) tends to 0 because the difference between \(a_n\) and \(b\) becomes arbitrarily small as \(n \to \infty\).  
   Therefore, \(\lim_{n \to \infty} |a_n - b| = 0\).

2. **Second statement**:  
   *Demostrar: Si la sucesión \(a_n\) converge a 0 y la sucesión \(b_n\) es acotada, entonces la sucesión \(a_n b_n\) también converge a 0.*  
   
   **Proof**:  
   Since \(a_n \to 0\), for every \(\epsilon > 0\), there exists \(N_1\) such that \(|a_n| < \epsilon/M\) for all \(n > N_1\), where \(M\) is the bound of the sequence \(b_n\), i.e., \(|b_n| \leq M\).  
   Now, consider the product \(|a_n b_n| = |a_n| |b_n|\).  
   For \(n > N_1\), we have \(|a_n b_n| \leq M|a_n| < M \cdot \epsilon/M = \epsilon\).  
   Hence, \(a_n b_n \to 0\).

3. **Third statement**:  
   *Demostrar: Si \(a_n \to a\) y \(b_n \to b\), entonces la sucesión \(\left(a_n + b_n\right)\) converge a \(a + b\).*  
   
   **Proof**:  
   Since \(a_n \to a\) and \(b_n \to b\), for any \(\epsilon > 0\), there exist \(N_1\) and \(N_2\) such that for \(n > N_1\), \(|a_n - a| < \epsilon/2\) and for \(n > N_2\), \(|b_n - b| < \epsilon/2\).  
   Let \(N = \max(N_1, N_2)\).  
   Then, for all \(n > N\), we have:
   \[
   |(a_n + b_n) - (a + b)| \leq |a_n - a| + |b_n - b| < \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon.
   \]
   Hence, \(\lim_{n \to \infty} (a_n + b_n) = a + b\).

4. **Fourth statement**:  
   *Demostrar: Si \(a_n \to a\) y \(b_n \to b\), entonces la sucesión \(a_n b_n\) converge a \(ab\).*  
   
   **Proof**:  
   Since \(a_n \to a\) and \(b_n \to b\), for any \(\epsilon > 0\), there exist \(N_1\) and \(N_2\) such that for \(n > N_1\), \(|a_n - a| < \epsilon/(2(|b| + 1))\) and for \(n > N_2\), \(|b_n - b| < \epsilon/(2(|a| + 1))\).  
   Let \(N = \max(N_1, N_2)\).  
   For all \(n > N\), we have:
   \[
   |a_n b_n - ab| = |a_n b_n - a b_n + a b_n - ab| \leq |a_n - a||b_n| + |b_n - b||a|.
   \]
   Since both terms tend to 0, the product \(a_n b_n \to ab\).

5. **Fifth statement**:  
   *Demostrar: Si \(a_n \to a\) y \(b_n \to b\), donde \(b_n \neq 0\) y \(b \neq 0\), entonces la sucesión \(\left(\frac{a_n}{b_n}\right)\) converge a \(\frac{a}{b}\).*  
   
   **Proof**:  
   Since \(a_n \to a\) and \(b_n \to b\), and \(b \neq 0\), there exists \(N_2\) such that for \(n > N_2\), \(b_n\) is bounded away from 0, i.e., \(|b_n| \geq c > 0\).  
   We know that:
   \[
   \frac{a_n}{b_n} - \frac{a}{b} = \frac{a_n b - a b_n}{b_n b} = \frac{(a_n - a)b + a(b - b_n)}{b_n b}.
   \]
   Since both \(a_n \to a\) and \(b_n \to b\), the numerator tends to 0, and the denominator remains bounded, so \(\frac{a_n}{b_n} \to \frac{a}{b}\).

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Would you like to discuss any particular proof in detail?  
Here are 5 related questions to expand this topic:

1. What happens if one of the sequences diverges? Does the result hold?
2. Can we prove convergence of sequences involving more complex functions like exponentials?
3. How does the proof change if the sequences oscillate instead of converging smoothly?
4. What is the role of boundedness in these proofs, especially in product and quotient cases?
5. Can these results be generalized for sequences of vectors in Euclidean space?

**Tip**: Always remember that the convergence of a product or quotient depends on both sequences involved. It's essential that the denominator does not approach zero for quotients.

Demostrar la convergencia de sucesiones dadas en las siguientes condiciones:

Explore the detailed proofs for the convergence of sequences under various conditions, including sums, products, and quotients. This material is suitable for university-level students learning about real analysis and sequence limits.

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