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Boundedness Theorem

• Statement:
  If $f_n : E \to \mathbb{R}$ is bounded on $E$ for every $n \in \mathbb{N}$, and $f_n \to f$ uniformly on $E$, then $f : E \to \mathbb{R}$ is also bounded on $E$.

Example

• Sequence of Functions:
  $f_n(x) = \frac{n}{nx + 1}$ for $x \in (0, 1)$.

• Observation:

  • Each $f_n$ is bounded on $(0, 1)$.
  • Pointwise Limit: $f(x) = \frac{1}{x}$, which is not bounded on $(0, 1)$.
  • Conclusion: The sequence $f_n$ cannot converge uniformly on $(0, 1)$.

Would you like more details or examples? Here are related questions you can explore:

1. What is the difference between pointwise and uniform convergence?
2. How can we formally prove the boundedness theorem?
3. Why is the function $f(x) = \frac{1}{x}$ unbounded on $(0, 1)$?
4. What are other examples where pointwise limits fail to be bounded?
5. How does the boundedness property relate to practical problems in analysis?

Tip: Uniform convergence ensures the limit function inherits properties like boundedness and continuity—pointwise convergence alone does not guarantee this!