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The image you've uploaded contains a problem related to a sequence of functions. The specific problem (part b) asks to **prove that \(\lim_{n \to \infty} v_n(x) = 0\) for all \(x \in \mathbb{R}\), where \(v_n(x)\) is defined as \(v_n(x) = u_n(x) - u(x)\)**.

### Given:
1. A sequence of functions \( \{u_n(x)\} \) converges pointwise to a function \( u(x) \).
2. \( v_n(x) = u_n(x) - u(x) \).

### To Prove:
\[
\lim_{n \to \infty} v_n(x) = 0 \text{ for all } x \in \mathbb{R}.
\]

### Proof:
1. **Pointwise Convergence Definition**: 
   By definition, \( u_n(x) \) converges pointwise to \( u(x) \) if for every \( x \in \mathbb{R} \) and for every \( \epsilon > 0 \), there exists a natural number \( N \) such that for all \( n \geq N \),
   \[
   |u_n(x) - u(x)| < \epsilon.
   \]
   This implies that \( \lim_{n \to \infty} u_n(x) = u(x) \).

2. **Relation Between \( v_n(x) \) and \( u_n(x) \)**:
   From the definition of \( v_n(x) \), we have:
   \[
   v_n(x) = u_n(x) - u(x).
   \]
   Therefore, 
   \[
   |v_n(x)| = |u_n(x) - u(x)|.
   \]
   By the pointwise convergence of \( u_n(x) \) to \( u(x) \), it follows that:
   \[
   \lim_{n \to \infty} |v_n(x)| = \lim_{n \to \infty} |u_n(x) - u(x)| = 0.
   \]

3. **Conclusion**:
   Since \( |v_n(x)| \) converges to \( 0 \) for every \( x \in \mathbb{R} \), we conclude that:
   \[
   \lim_{n \to \infty} v_n(x) = 0 \text{ for all } x \in \mathbb{R}.
   \]

Thus, the proof is complete.

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Would you like more details or have any further questions?

Here are some related questions you might consider:
1. What is the difference between pointwise and uniform convergence?
2. How does the concept of uniform convergence relate to the convergence of sequences of functions?
3. Can pointwise convergence guarantee continuity of the limit function?
4. What is an example of a sequence of functions that converges pointwise but not uniformly?
5. How do we prove the convergence of a sequence of functions using the Cauchy criterion?

**Tip:** When working with sequences of functions, always check if the convergence is pointwise or uniform, as this can affect the properties of the limit function.

Learn how to prove that the limit of the sequence of functions v_n(x) = u_n(x) - u(x) converges to 0 for all x in real numbers. This problem explores pointwise convergence and the limit of functions, suitable for advanced undergraduate level mathematics.

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