Question

(a) Show that if \(\left\\{F_{n}\right\\}\) converges uniformly on \(S,\) then \(\left\\{F_{n}\right\\}\) converges uniformly on every subset of \(S\). (b) Show that if \(\left\\{F_{n}\right\\}\) converges uniformly on \(S_{1}, S_{2}, \ldots, S_{m},\) then \(\left\\{F_{n}\right\\}\) converges uniformly on \(\bigcup_{k=1}^{m} S_{k}\). (c) Give an example where \(\left\\{F_{n}\right\\}\) converges uniformly on each of an infinite sequence of sets \(S_{1}, S_{2}, \ldots,\) but not on \(\bigcup_{k=1}^{\infty} S_{k}\).

Short answer

In summary, we have shown that if a sequence of functions converges uniformly on a set S, it also converges uniformly on every subset of S. Additionally, if a sequence of functions converges uniformly on sets \(S_1, S_2, \ldots, S_m\), it will also converge uniformly on the union of these sets. However, we provided an example demonstrating that uniform convergence on each member of an infinite sequence of sets does not guarantee uniform convergence on the infinite union of these sets.

Step-by-step solution

Define Uniform Convergence

Recall that a sequence of functions \(\left\{F_{n}\right\}\) converges uniformly to a function F on a set S if, for every \(\varepsilon > 0\), there exists a natural number \(N\) such that, for all \(n \geq N\) and for all \(x \in S\), we have \(|F_n(x) - F(x)| < \varepsilon\).

Pick an arbitrary subset and apply the definition

Let \(T\) be an arbitrary subset of \(S\). We want to show that \(\left\{F_{n}\right\}\) converges uniformly to F on \(T\). Let \(\varepsilon > 0\). Since \(\left\{F_{n}\right\}\) converges uniformly on \(S\), there exists a natural number \(N\) such that, for all \(n \geq N\) and for all \(x \in S\), we have \(|F_n(x) - F(x)| < \varepsilon\).

Show uniform convergence on the subset

Now, let \(x \in T\). Since \(T\) is a subset of \(S\), \(x \in S\). Hence, for all \(n \geq N\) and for all \(x \in T\), we have \(|F_n(x) - F(x)| < \varepsilon\). This shows that \(\left\{F_{n}\right\}\) converges uniformly on every subset \(T\) of \(S\). (b) Convergence on the Union

Define the union of sets

Let \(S = \bigcup_{k=1}^{m} S_{k}\) be the union of sets \(S_1, S_2, \ldots, S_m\).

Apply the definition of uniform convergence

We want to show that \(\left\{F_{n}\right\}\) converges uniformly on \(S\). Let \(\varepsilon > 0\). Since \(\left\{F_{n}\right\}\) converges uniformly on \(S_k\) for each \(k = 1, 2, \ldots, m\), there exists a natural number \(N_k\) such that, for all \(n \geq N_k\) and for all \(x \in S_k\), we have \(|F_n(x) - F(x)| < \varepsilon\).

Choose an N that works for all sets

Let \(N = \max\{N_1, N_2, \ldots, N_m\}\). Then, for all \(n \geq N\) and for all \(x \in S_k\) (for each \(k\)), we have \(|F_n(x) - F(x)| < \varepsilon\). Since \(S = \bigcup_{k=1}^{m} S_{k}\), this implies that, for all \(n \geq N\) and for all \(x \in S\), we have \(|F_n(x) - F(x)| < \varepsilon\). Hence, \(\left\{F_{n}\right\}\) converges uniformly on the union \(S = \bigcup_{k=1}^{m} S_{k}\). (c) Example with Convergence on Infinite Sequence of Sets

Define a sequence of functions

Let \(F_n(x) = x^n\) be a sequence of functions defined on the interval \((0, 1)\).

Establish convergence on an infinite sequence of sets

Let \(S_k = (\frac{1}{k}, 1-\frac{1}{k})\) for \(k \in \mathbb{N}\). Clearly, \(F_n(x) \rightarrow 0\) as \(n \rightarrow \infty\) for all \(x \in S_k\). Furthermore, for any fixed \(x\) in \(S_k\), the functions \(F_n(x)\) converge uniformly to 0 since the convergence rate does not depend on the choice of \(x\).

Show non-convergence on the infinite union

Now we consider the infinite union \(S = \bigcup_{k=1}^{\infty} S_{k} = (0, 1)\). For any \(x \in (0, 1)\) and \(n \in \mathbb{N}\), \(0 \leq F_n(x) \leq 1\). However, since \(x^n \rightarrow 0\) as \(n \rightarrow \infty\) for any \(x \in (0, 1)\), the functions \(F_n(x)\) converges pointwise to 0 on \((0, 1)\) but does not converge uniformly, as the convergence rate at the endpoints 0 and 1 is significantly slower. That is, the required \(N\) in the definition of uniform convergence would depend on the choice of \(x\) on this interval. This example shows that a sequence of functions can converge uniformly on each of an infinite sequence of sets such as \(S_1, S_2, \ldots,\) but not on their infinite union \(\bigcup_{k=1}^{\infty}S_k\).

Key concepts

Sequence of Functions

Understanding the behavior of a sequence of functions is fundamental for various fields in mathematics and applications. When we consider a sequence of functions, \( \{F_n(x)\} \), it is a collection of functions that depend on a natural number \( n \). Each function in the sequence may be defined on the same domain and take values within the same codomain.

As \( n \) increases, we observe how the functions \( F_n(x) \) behave for every point \( x \) in the domain. The key question that arises with a sequence of functions is whether they approach a well-defined limit function as \( n \) goes to infinity. If there is such a limit function, \( F(x) \), and if the sequence \( F_n(x) \) approaches \( F(x) \) 'nicely enough' (in a way we will discuss under uniform convergence), then we say that the sequence \( F_n(x) \) converges to \( F(x) \).

However, not all sequences of functions converge, and the manner of convergence is crucial. If they do converge, we must consider various modes of convergence such as pointwise, uniform, or in some metric space sense. Each of these offers unique insights into the structure and characteristics of the sequence.

Convergence of Functions

The mode of convergence for a sequence of functions reveals much about its behavior, and uniform convergence is particularly significant because of its strong implications. For a sequence \( \{F_n\} \) of functions to converge uniformly to a function \( F \) on a set \( S \) signifies that the functions \( F_n \) not only approach \( F \) but do so at a rate that does not depend on the point \( x \) in \( S \)—a very stringent requirement.

This form of convergence is vital because it preserves many properties between the functions in the sequence and the limit function, such as continuity and integrability. For example, if each function \( F_n \) is continuous and the sequence converges uniformly to \( F \), then \( F \) will also be continuous.

Uniform convergence, as opposed to pointwise convergence, ensures that for any chosen precision \( \varepsilon > 0 \), there exists an \( N \) such that for all \( n \geq N \) and for every \( x \) in \( S \) the inequality \( |F_n(x) - F(x)| < \varepsilon \) holds. It implies that the closeness of \( F_n \) to \( F \) is uniform across the entire set \( S \) and is not affected by the specific values that \( x \) might take.

Infinite Sequence of Sets

When pondering convergence over an infinite sequence of sets, things get a bit tricky. An infinite sequence of sets \( S_1, S_2, \ldots \) suggests that we have a countably infinite collection of sets often related in their construction or context.

For each finite subset of this sequence, a sequence of functions may well converge uniformly. However, when we consider the infinite union of these sets, the uniform convergence on each individual set does not immediately guarantee uniform convergence on the combined set. The core issue hinges on the fact that the conditions for uniform convergence must hold uniformly over the entire domain, and as the union extends to include more sets, it becomes increasingly difficult to satisfy this stringent requirement because the nature of the problem may change fundamentally as the set grows.

An instructive example shows that uniform convergence on each set inside an infinite sequence does not extend to their infinite union. This is a crucial concept that highlights the delicate nature of uniform convergence and the careful analysis required when extending results from finite to infinite contexts. It underscores the importance of understanding the unique properties of convergence with respect to both the sequence of functions and the domain over which it is considered.