Question

Prove that if \(f_{n} \rightarrow f\) (uniformly) on \(B\) and if \(C \subseteq B,\) then \(f_{n} \rightarrow f\) (uniformly) on \(C\) as well.

Short answer

Uniform convergence on a set implies uniform convergence on any subset of that set.

Step-by-step solution

Understanding Uniform Convergence

Uniform convergence of a sequence of functions \(f_n\) to a function \(f\) on a set \(B\) means that for every \(\epsilon > 0\), there exists an \(N\) such that for all \(n > N\) and for all \(x \in B\), \(|f_n(x) - f(x)| < \epsilon\). This definition implies that the convergence is independent of the specific point \(x\) in the set \(B\).

Impose Condition on Subset

Consider a subset \(C\) of \(B\). We need to show that \(f_n\) converges uniformly to \(f\) on \(C\) as well. Since \(C \subseteq B\), any point \(x \in C\) is also in \(B\).

Apply Existing Condition from Uniform Convergence

Since \(f_n\) converges uniformly to \(f\) on \(B\), for every \(\epsilon > 0\), there is an \(N\) such that for every \(n > N\) and all \(x \in B\), we have \(|f_n(x) - f(x)| < \epsilon\). This automatically includes all points \(x \in C\) because \(C \subseteq B\).

Conclude Uniform Convergence on Subset

Given that the condition \(|f_n(x) - f(x)| < \epsilon\) holds for all \(x \in B\) and specifically for all \(x \in C\), it follows that \(f_n\) converges uniformly to \(f\) on \(C\). Thus, the uniform convergence on the larger set \(B\) directly implies uniform convergence on its subset \(C\).

Key concepts

Sequence of Functions

A sequence of functions involves a list of functions, each with its own domain and range, indexed by natural numbers. Consider this like a line at a movie theater with each person (or function) holding a ticket to a specific movie (or value). As we move down the line, each function consequently takes its turn.
In mathematical terms, it’s often expressed as \(f_1, f_2, f_3, \ldots, f_n\). Each \(f_n\) is a function and all of these form a sequence.
Understanding this sequence is critical when talking about convergence. Convergence tells us how the functions behave as \(n\) goes to infinity.

• Each function maps elements from a domain to a co-domain.
• The idea of convergence focuses on what happens as the sequence extends indefinitely.
• Uniform convergence ensures convergence of the sequence at every point of a given set.

These aspects of sequences are essential when dealing with multiple functions approaching a limit function.

Subset

A subset in mathematics is essentially a smaller collection of elements taken from a larger collection, known as a set. Visualize this by picturing a group of students (set) and selecting some students wearing red shirts (subset).
In our context, we are given that \(C\) is a subset of \(B\). This means every element \(x\) that is in \(C\) is also an element of \(B\).
Subsets play a crucial role in examining how certain properties, such as convergence, behave not only within an entire set but also within portions of it.

• Subsets can be thought of as portions of larger sets.
• This helps when extending properties such as uniform convergence from the larger set \(B\) to its subset \(C\).
• Understanding subsets ensures we can talk about specific elements within a larger group without losing generality.

This understanding is key when discussing the behavior of functions over specific domains.

Epsilon-Delta Definition

The epsilon-delta definition is a formal way of defining the concept of limits and convergence in calculus. This involves two constants, \(\epsilon\) which represents an arbitrarily small number, and \(\delta\) associated with an interval or distance.
In terms of function sequences, uniform convergence means for every \(\epsilon > 0\), there is a corresponding \(N\) such that if \(n > N\), then \(|f_n(x) - f(x)| < \epsilon\) for all \(x\) in the domain.

• \(\epsilon\) is used to express how close we want to be to our limit.
• \(\delta\) is used to define the proximity to a particular value beyond which all elements must lie.
• This forms the backbone of the rigorous proof concept in calculus.

The epsilon-delta definition assures us how diverse elements within a set behave uniformly, mirroring the core ideas behind uniform convergence.

Mathematical Proof

A mathematical proof is a logical argument demonstrating the truth of a statement, constructed using accepted mathematical principles. Proofs ensure that mathematical concepts are free from doubt.
In demonstrating uniform convergence, we apply this reasoning by using the properties of the function sequence and subsets to show consistent behavior.
In our example, we use proof to show that if every function \(f_n\) in a sequence converges uniformly on a set \(B\), then it will also converge on any subset \(C\) of \(B\).

• Proofs often start with known conditions and definitions, like the epsilon-delta condition for uniform convergence.
• They explore relationships logically and determine consistency with established results.
• Proofs in calculus require methodical steps to demonstrate that sequences maintain limits uniformly on subsets.

Proofs in mathematics are the stepping stones that enable us to solidify our understanding and assertions about complex concepts like sequences and convergence.