Question

Show that if \(x_{n} \rightarrow x\) in a normed vector space then $$ \frac{x_{1}+x_{2}+\cdots+x_{n}}{n} \rightarrow x $$

Short answer

The key point is to express the difference between the arithmetic mean and x, and make each term less than \( \frac{\varepsilon}{2} \) by carefully choosing N and M based on the definition of sequence convergence in a normed vector space. The steps of the proof show that both terms can be made arbitrarily small, thus the arithmetic mean of the sequence converges to x.

Step-by-step solution

Definition of sequence convergence in a normed vector space

Given a sequence \( x_n \) in a normed vector space that converges to x, this means that for any \( \varepsilon > 0 \), there exists a natural number N such that for all \( n > N \), \( ||x_n - x|| < \varepsilon \).

Define the arithmetic mean

The arithmetic mean of the sequence \( x_n \) up to the nth term is defined as \( \frac{x_1 + x_2 + \cdots + x_n}{n} \). In order to prove that the arithmetic mean converges to x, we need to show that for any \( \varepsilon > 0 \), there exists a natural number M such that for all \( n > M \) we have \( ||\frac{x_1 + x_2 + \cdots + x_n}{n} - x|| < \varepsilon \).

Express the norm of the arithmetic mean minus x

Consider \( ||\frac{x_1 + x_2 + \cdots + x_n}{n} - x|| \). This can be rewritten as \( ||\frac{x_1 + x_2 + \cdots + x_N + x_{N + 1} + \cdots + x_n}{n} - x|| \). Expand this and apply triangle inequality to obtain \( ||\frac{x_1 + x_2 + \cdots + x_N}{n} - \frac{Nx}{n} + \frac{x_{N + 1} + \cdots + x_n}{n} - \frac{(n - N)x}{n}|| \leq \frac{1}{n}||x_1 + x_2 + \cdots + x_N - Nx|| + \frac{1}{n}||x_{N + 1} + \cdots + x_n - (n - N)x||.

Apply the convergence definition of \( x_n \)

Since \( x_n \rightarrow x \) for \( n > N \), by choosing \( N > \frac{2}{\varepsilon}||x_1 + x_2 + \cdots + x_N - Nx|| \) the first term of the inequality in the previous step can be made less than \( \frac{\varepsilon}{2} \). For the second term, choose \( M > \max {N} \) such that for \( n > M \) and for \( n > N \), we have \( ||x_n - x|| < \frac{\varepsilon}{2(n - N)} \). Therefore, the second term can also be made less than \( \frac{\varepsilon}{2} \).

Conclude the proof

Since both terms from the inequality of step 3 can be made less than \( \frac{\varepsilon}{2} \), we then have \( ||\frac{x_1 + x_2 + \cdots + x_n}{n} - x|| = ||mean_{n=1}^{N}{x_n} - x|| < \varepsilon \). This means that as \( n \) gets very large, the means \( mean_{n=1}^{n}{x_n} \) converge to \( x \), hence the proof is completed.

Key concepts

Sequence Convergence

In mathematical analysis, sequence convergence is a fundamental concept. It describes the behavior of a sequence as its terms approach a particular value, known as the limit, as the sequence progresses. For a sequence \(x_n\) in a normed vector space to converge to a point \(x\), for every positive number \(\varepsilon\), there must be a natural number \(N\) such that for all \(n > N\), the expression \(||x_n - x|| < \varepsilon\) holds true.
This means that the distance between \(x_n\) and \(x\) becomes arbitrarily small as \(n\) increases. In essence, convergence ensures that the sequence gets closer and closer to its limit.
This concept is pivotal in understanding how sequences behave within the framework of normed vector spaces.

Arithmetic Mean

The arithmetic mean, often simply called the average, is calculated by summing a set of numbers and dividing by the quantity of those numbers. In the context of sequences, we define it as \(\frac{x_1 + x_2 + \cdots + x_n}{n}\). This mean represents a sequence of averages as we take more terms into consideration.

• The primary question is whether these averages converge to the same limit as the original sequence \(x_n\).
• To prove convergence of arithmetic means, it is essential to show that for any small positive number \(\varepsilon\), there exists an \(M\) such that for every \(n > M\), \( ||\frac{x_1 + x_2 + \cdots + x_n}{n} - x|| < \varepsilon \).

This approach ensures that the average values converge towards the same limit \(x\) as the sequence.

Triangle Inequality

The triangle inequality is a critical tool in analyzing sequences in normed vector spaces. This principle states that for any vector norm, \(||a + b|| \leq ||a|| + ||b||\).
It plays a vital role when dealing with the expression for the arithmetic mean minus the limit \(x\). Applying this inequality allows us to bound the norm by splitting it into components that can be individually assessed and controlled.
For example, in proving the convergence of means, the triangle inequality helps to decompose the norm difference between the sequence and the mean into parts that can both be made less than \(\varepsilon/2\). This step is crucial as it makes the proof feasible by letting us manage each segment independently and combine their bounds effectively.

Normed Vector Space

A normed vector space builds on vector spaces but introduces an additional structure through norms. A norm is a function that assigns a non-negative length or size to each vector in the space, symbolized as \(||x||\).

• Norms must satisfy properties such as non-negativity \(||x|| \geq 0\), the triangle inequality \(||x + y|| \leq ||x|| + ||y||\), and scaling \(||\lambda x|| = |\lambda| \cdot ||x||\).
• These properties provide a rigorous framework to analyze vector behavior and measure possibilities within the space.

By equipping vector spaces with norms, it is possible to rigorously discuss concepts like convergence, continuity, and boundedness. This foundation is essential to exploring deeper mathematical theories and principles.

Epsilon-Delta Definition

The epsilon-delta definition is a cornerstone in understanding limits and continuity in mathematics. For sequences, it provides a way to describe when a sequence converges to a limit. According to this definition, \(x_n\) converges to \(x\) if, for every \(\varepsilon > 0\), there exists a natural number \(N\) such that for every \(n > N\), \(||x_n - x|| < \varepsilon\).

• This epsilon-delta framework ensures that the terms of the sequence \(x_n\) are within an \(\varepsilon\) distance from \(x\) for all sufficiently large \(n\).
• It's a precise method to encapsulate the intuitive notion of getting closer and closer.

This rigorous approach allows mathematicians to make conclusive statements about the behavior of sequences and their tendency towards a specific value.