Define
fn(x) :=
(
0 if n ≤ x ≤ n + 1
1 otherwise.
Show that the sequence fn converges to 1 pointwise, but not uniformly.

We define the sequence of functions \( f_n: \mathbb{R} \to \mathbb{R} \) by

\[
f_n(x) =
\begin{cases}
0, & \text{if } n \leq x \leq n+1, \\
1, & \text{otherwise}.
\end{cases}
\]

### **Step 1: Pointwise Convergence**
We say that \( f_n \) converges pointwise to a function \( f(x) \) if for every fixed \( x \in \mathbb{R} \),

\[
\lim_{n \to \infty} f_n(x) = f(x).
\]

- If \( x \) is **fixed**, then there are two cases:
  1. If \( x \in [n, n+1] \) for some \( n \), then \( f_n(x) = 0 \). However, for sufficiently large \( n \), \( x \) will no longer be in the interval \( [n, n+1] \), meaning that for large \( n \), we have \( f_n(x) = 1 \).
  2. If \( x \notin [n, n+1] \) for all large \( n \), then \( f_n(x) = 1 \) for all sufficiently large \( n \).

- In both cases, we see that

\[
\lim_{n \to \infty} f_n(x) = 1.
\]

Thus, the pointwise limit of \( f_n(x) \) is the constant function:

\[
f(x) = 1, \quad \forall x \in \mathbb{R}.
\]

### **Step 2: Uniform Convergence**
A sequence of functions \( f_n \) converges **uniformly** to \( f \) if

\[
\sup_{x \in \mathbb{R}} |f_n(x) - f(x)| \to 0 \quad \text{as } n \to \infty.
\]

Here, we compute:

\[
\sup_{x \in \mathbb{R}} |f_n(x) - 1|.
\]

Since \( f(x) = 1 \) for all \( x \), we find:

\[
|f_n(x) - 1| =
\begin{cases}
|0 - 1| = 1, & \text{if } n \leq x \leq n+1, \\
|1 - 1| = 0, & \text{otherwise}.
\end{cases}
\]

Thus, the **supremum** is:

\[
\sup_{x \in \mathbb{R}} |f_n(x) - 1| = 1.
\]

Since this does not tend to \( 0 \) as \( n \to \infty \), the sequence \( f_n(x) \) **does not** converge uniformly to \( f(x) \).

### **Conclusion**
We have shown that:

1. \( f_n(x) \to 1 \) **pointwise** for all \( x \).
2. \( f_n(x) \) **does not** converge uniformly to \( 1 \) because the supremum of the deviation remains 1.

Thus, the sequence \( f_n(x) \) converges to \( 1 \) **pointwise but not uniformly**.

---

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### **Related Questions**
1. What is the difference between pointwise and uniform convergence?
2. How does the definition of uniform convergence ensure uniform closeness to the limit function?
3. Can you provide another example of a sequence of functions that converges pointwise but not uniformly?
4. What are some sufficient conditions for uniform convergence of a sequence of functions?
5. How does the uniform norm help in determining uniform convergence?

**Tip:** Uniform convergence is stronger than pointwise convergence because it ensures that the functions converge uniformly over the entire domain, making it possible to interchange limits with integration and differentiation.

Define fn(x) := (0 if n ≤ x ≤ n + 1, 1 otherwise. Show that the sequence fn converges to 1 pointwise, but not uniformly.

Learn how the sequence of functions fn(x) defined by 0 if n ≤ x ≤ n + 1 and 1 otherwise converges to 1 pointwise but not uniformly. This detailed solution demonstrates the concepts of pointwise and uniform convergence with key theorems and step-by-step reasoning. Suitable for high school students (Grades 11-12).

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