Math Problem Statement
- Prove that every uniformly convergent sequence of bounded functions is uniformly bounded.
Solution
Let be a sequence of functions defined on a set . We are given that the sequence converges uniformly to some function , i.e.,
Additionally, we are told that each function is bounded. That is, for each , there exists a constant such that
Goal:
We aim to prove that the sequence is uniformly bounded, which means that there exists a constant such that
Proof:
- Uniform Convergence: By the definition of uniform convergence, since uniformly, for any , there exists an integer such that for all and all ,
- Boundedness of : Since each is bounded, there exists a constant such that for all ,
Thus, for each , we know that it has some bound for every , but the goal is to show that we can find a common bound that works for all .
- Finding a Uniform Bound: Since uniformly, we know that as becomes large, the functions get arbitrarily close to . Thus, for , we have
Now, define
Since each is bounded, the supremum exists and is finite. Moreover, by the uniform convergence of to , we can say that the sequence of bounds does not grow arbitrarily large because the uniform convergence ensures that the terms stay close to for large .
Thus, we conclude that the sequence is uniformly bounded.
Conclusion:
The sequence of uniformly convergent functions is uniformly bounded, i.e., there exists a constant such that
Would you like more details on any part of the proof? Here are some related questions for further exploration:
- What does uniform convergence imply about the relationship between the pointwise limit and the sequence of functions?
- How can we prove that a pointwise convergent sequence of bounded functions is not necessarily uniformly bounded?
- What is the difference between uniform convergence and pointwise convergence?
- Can you show an example where a sequence of functions is uniformly bounded but not uniformly convergent?
- How does the concept of uniform convergence relate to the Weierstrass M-test?
Tip: To prove that a sequence of functions is uniformly bounded, try to show that the bounds do not depend on the particular choice of in the domain, but rather hold for all .
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Math Problem Analysis
Mathematical Concepts
Uniform Convergence
Bounded Functions
Supremum
Limits
Formulas
lim_{n \to \infty} sup_{x \in X} |f_n(x) - f(x)| = 0
|f_n(x)| \leq M_n
|f_n(x)| \leq M + |f(x)|
Theorems
Weierstrass M-test
Uniform Convergence Theorem
Suitable Grade Level
Undergraduate Level (Mathematics)
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