Question

Prove that if \(x_{n} \leq y_{n}\) for all \(n\), then \(\underline{\lim } x_{n} \leq \underline{\lim } y_{n}\) and \(\lim x_{n} \leq \overline{\lim } y_{n}\).

Short answer

Given \(x_n \leq y_n\) for all \(n\), it follows that \( \underline{\lim } x_n \leq \underline{\lim } y_n \) and \( \lim x_n \leq \overline{\lim } y_n \).

Step-by-step solution

Identify Limsup and Liminf

First, understand the notation: - \( \underline{\lim} x_n \) represents the limit inferior of the sequence \( x_n \).- \( \overline{\lim} y_n \) represents the limit superior of the sequence \( y_n \).The limit inferior is the greatest lower bound of the set of subsequential limits, while the limit superior is the least upper bound of the set of subsequential limits. Our goal is to show a comparison between these bounds for \( x_n \) and \( y_n \).

Establish Sequence Inequality

We start with the given condition that \( x_n \leq y_n \) for all \( n \). This implies that any subsequence of \( x_n \) is also less than or equal to the corresponding subsequence of \( y_n \), preserving the inequality across subsequences.

Proof for Limit Inferior

Since \( x_n \leq y_n \) for all \( n \), each subsequential limit of \( x_n \) will also be less than or equal to the corresponding subsequential limit of \( y_n \). Thus, the greatest lower bound (limit inferior) of the \( x_n \) sequence will be less than or equal to the greatest lower bound of the \( y_n \) sequence, hence \( \underline{\lim } x_{n} \leq \underline{\lim } y_{n} \).

Proof for Traditional Limit and Limit Superior

For the traditional limit sequential analysis, suppose \( \lim x_n = L \), then by the sequence inequality, \( L \leq \overline{\lim } y_n \) because \( L \) is also a limit inferior. Also, any limit point \( L \) of \( x_n \) must be less than or equal to all limit points of \( y_n \), reinforcing \( L \leq \overline{\lim } y_n \).

Conclusion

By arguing through the properties of subsequences and the bounding definitions of limsup and liminf, we conclude that:- \( \underline{\lim } x_n \leq \underline{\lim } y_n \)- \( \lim x_n \leq \overline{\lim } y_n \).These relations hold true under the original inequality constraint.

Key concepts

Limit Inferior

The limit inferior, often denoted as uline{\lim} \ x_n, refers to the greatest lower bound of the subsequential limits within a sequence. Conceptually, it finds a value that all smaller subsequential limits shy away from, acting like a baseline where values begin to cluster. Think of it as the minimum level you can reasonably expect subsequences to reach consistently, without dipping lower over time.

Understanding limit inferior can be crucial when analyzing bounded sequences, where certain behaviors ensure the existence of subsequential limits. For any sequence \( x_n \), the limit inferior lies above every possible minimum limit point, providing a comprehensive view over time. Despite fluctuations in a sequence, limit inferior remains a trustworthy benchmark.

In the context of inequalities, as proved in exercises, if \( x_n \leq y_n \) for each \( n \), then \( \uline{\lim} x_n \leq \uline{\lim} y_n \). This maintains the integrity of sequence inequality, preserving the order of limit inferior values as an inherent quality.

Limit Superior

Limit superior, expressed as \( \oline{\lim} \ y_n \), is essentially the opposite of the limit inferior. It signifies the least upper bound of all subsequential limits in a sequence. This concept highlights the peak, or the highest values that subsequential limits persistently achieve over the duration of the sequence.

Identifying limit superior in sequences allows for clear visualization of potential maximum boundaries that won't be exceeded. It provides an understanding of what upper thresholds sequences tend to converge around as time passes. Limit superior is particularly beneficial for determining the endurance and feasibility of reaching higher values within fluctuating data.

Given \( x_n \leq y_n \) for all \( n \), it's established that \( \lim x_n \leq \oline{\lim} y_n \). Basically, it ensures that the highest thresholds \( y_n \) can reach respect the upper constraints set by \( x_n \), showing that subsequential tendencies adhere to initial constraints.

Subsequential Limits

Subsequential limits are derived from subsequences of a larger sequence and represent potential limit points this sequence can approach. This involves breaking down a sequence into various subsequences (sub-parts maintaining original order without gaps), each possessing its own limit. Understanding subsequential limits is like scrutinizing a collection of mini-paths within an overarching trajectory, each offering insights into how the original sequence behaves under different conditions or perspectives.

Why are they significant? Because they reveal all possible stabilization points of a sequence. When analyzing sequences through subsequential limits, one can ascertain whether a sequence is bounded, or if it will converge to particular values or oscillate indefinitely. Thus, examining subsequential limits lends vital clarity to sequences that would otherwise appear chaotic or boundless.

In the exercise, subsequential limits play a key role in comparing sequence bounds to prove inequalities such as \( \uline{\lim} x_n \leq \uline{\lim} y_n \) and \( \lim x_n \leq \oline{\lim} y_n \). This is because each subsequential limit of one sequence must still conform to the inequality constraints posed by the subsequences of the other.