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The problem in the image asks to demonstrate that the sequence of functions \( \{ f_n(x) \}_n \), defined as

\[
f_n(x) = 
\begin{cases}
\frac{x}{n}, & \text{if } x \in [0,1], \\
0, & \text{otherwise},
\end{cases}
\]
is pointwise convergent and converges to the function \( f(x) = 0 \) for all \( x \).

### Solution:

#### 1. Definition of Pointwise Convergence:
A sequence of functions \( f_n(x) \) converges pointwise to a function \( f(x) \) on a domain \( D \) if, for every \( x \in D \), 
\[
\lim_{n \to \infty} f_n(x) = f(x).
\]
In this case, we are asked to show that \( f_n(x) \to 0 \) for all \( x \).

#### 2. Behavior of \( f_n(x) \):
The function \( f_n(x) = \frac{x}{n} \) is defined for \( x \in [0, 1] \) and is 0 otherwise. We can analyze the pointwise limit by considering different cases for \( x \).

- **For \( x = 0 \)**:
  \[
  f_n(0) = \frac{0}{n} = 0 \quad \text{for all } n.
  \]
  Therefore, \( f_n(0) = 0 \) for all \( n \), so the pointwise limit is \( f(0) = 0 \).

- **For \( x \in (0,1] \)**:
  \[
  f_n(x) = \frac{x}{n}.
  \]
  As \( n \to \infty \), we have
  \[
  \lim_{n \to \infty} \frac{x}{n} = 0.
  \]
  Therefore, for any \( x \in (0, 1] \), \( f_n(x) \to 0 \) as \( n \to \infty \).

- **For \( x \notin [0,1] \)**:
  The function is defined as \( f_n(x) = 0 \) for all \( n \), so the limit is trivially \( f(x) = 0 \).

#### 3. Conclusion:
Thus, the sequence \( f_n(x) \) converges pointwise to the function \( f(x) = 0 \) for all \( x \in \mathbb{R} \).

Would you like further details on any of the steps?

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Here are 5 relative questions for further exploration:
1. How does uniform convergence differ from pointwise convergence?
2. What are some practical applications of pointwise convergence in real analysis?
3. Can we conclude uniform convergence in this case? Why or why not?
4. How do we extend this concept of convergence to other types of function sequences?
5. What are examples of function sequences that converge uniformly but not pointwise?

**Tip:** Always verify the domain of the functions and the limit to avoid confusion between pointwise and uniform convergence.

Consider the sequence of functions \( \{ f_n(x) \}_n \), defined as \( f_n(x) = \frac{x}{n} \) for \( x \in [0,1] \) and 0 otherwise. Show that this sequence of functions is pointwise convergent and converges to the function \( f(x) = 0 \) for all \( x \).

Explore how the sequence of functions \( f_n(x) = \frac{x}{n} \) for \( x \in [0,1] \) converges pointwise to the function \( f(x) = 0 \). This analysis covers key real analysis concepts, including pointwise convergence and sequences of functions. Suitable for undergraduate mathematics students.

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