Question

Assume that \(\bar{s}, \underline{s}\) ( or \(s\) ), \(\bar{t},\) and \(\underline{t}\) are in the extended reals, and show that the given inequalities or equations hold whenever their right sides are defined (not indeterminate). (a) If \(s_{n} \geq 0, t_{n} \geq 0,\) then (i) \(\varlimsup_{n \rightarrow \infty} s_{n} t_{n} \leq \bar{s} t\) and (ii) \(\varliminf_{n \rightarrow \infty} s_{n} t_{n} \geq \underline{s t}\). (b) If \(s_{n} \leq 0, t_{n} \geq 0,\) then (i) \(\varlimsup_{n \rightarrow \infty} s_{n} t_{n} \leq \bar{s} \underline{t}\) and (ii) \(\varliminf_{n \rightarrow \infty} s_{n} t_{n} \geq s \bar{t}\).

Short answer

Based on the given properties and conditions of the sequences \((s_n)\) and \((t_n)\), we proved four inequalities: (a) If \(s_n \geq 0\), \(t_n \geq 0\), then (i) \(\varlimsup_{n \rightarrow \infty} s_{n} t_{n} \leq \bar{s} t\) (ii) \(\varliminf_{n \rightarrow \infty} s_{n} t_{n} \geq \underline{s t}\) (b) If \(s_n \leq 0\), \(t_n \geq 0\), then (i) \(\varlimsup_{n \rightarrow \infty} s_{n} t_{n} \leq \bar{s} \underline{t}\) (ii) \(\varliminf_{n \rightarrow \infty} s_{n} t_{n} \geq s \bar{t}\) The process of proving these inequalities involved understanding the sequences' conditions and exploring the limit superior and limit inferior of the sequences. We used these understandings to analyze the respective inequalities and draw conclusions in each case.

Step-by-step solution

Understanding the condition \(s_n \geq 0\) and \(t_n \geq 0\)

Given that \(s_n \geq 0\) and \(t_n \geq 0\), this means that all the terms in the sequences \((s_{n})\) and \((t_{n})\) are non-negative real numbers.

Exploring the limit superior and limit inferior of a sequence

The limit superior of a sequence is the largest limit point of the sequence, while the limit inferior is the smallest limit point. Intuitively, \(\varlimsup_{n\rightarrow \infty} s_n\) represents the upper bound of the sequence \((s_{n})\), and similarly, \(\varliminf_{n\rightarrow \infty} s_n\) represents the lower bound of the sequence \((s_{n})\).

Prove the inequality (i) in (a)\(\varlimsup_{n \rightarrow \infty} s_{n} t_{n} \leq \bar{s} t\)

For any fixed positive integer \(n\), since \(s_n \geq 0\) and \(t_n \geq 0\), we have: \(s_{n}t_{n} \leq \bar{s}t_n\) because \(\bar{s}\) is the limit superior of \((s_n)\), and it's an upper bound for the sequence. Now taking the limit superior of both sides as \(n\rightarrow \infty\), using the property that the limit superior of a product is less than or equal to the product of the limit superiors (provided both sequences are non-negative), we get: \(\varlimsup_{n \rightarrow \infty} s_{n} t_{n} \leq \bar{s} \left(\varlimsup_{n \rightarrow \infty} t_n \right) = \bar{s} t\).

Proving the inequality (ii) in (a) \(\varliminf_{n \rightarrow \infty} s_{n} t_{n} \geq \underline{s t}\)

Since both \(s_n\) and \(t_n\) are non-negative, for any fixed \(n\), we have: \(s_{n}t_{n} \geq \underline{s}t_n\) because \(\underline{s}\) is the limit inferior of \((s_n)\), and it's a lower bound for the sequence. Now taking the limit inferior of both sides as \(n\rightarrow \infty\), using the property that the limit inferior of a product is greater than or equal to the product of limit inferiors (provided both sequences are non-negative), we get: \(\varliminf_{n \rightarrow \infty} s_{n} t_{n} \geq \underline{s} \left(\varliminf_{n \rightarrow \infty} t_n \right) = \underline{s} t\). Proof of (b):

Understanding the condition \(s_n \leq 0\) and \(t_n \geq 0\)

Given that \(s_n \leq 0\) and \(t_n \geq 0\), this means that all the terms in the sequence \((s_{n})\) are non-positive real numbers and terms in the sequence \((t_{n})\) are non-negative real numbers.

Prove the inequality (i) in (b) \(\varlimsup_{n \rightarrow \infty} s_{n} t_{n} \leq \bar{s} \underline{t}\)

For any fixed positive integer \(n\), since \(s_n \leq 0\) and \(t_n \geq 0\), we have: \(s_{n}t_{n} \leq \bar{s}\underline{t}\) because \(\bar{s}\) is the limit superior of \((s_n)\), and \(\underline{t}\) is the limit inferior of \((t_n)\). Now taking the limit superior of both sides as \(n\rightarrow \infty\), we get: \(\varlimsup_{n \rightarrow \infty} s_{n} t_{n} \leq \bar{s}\underline{t}\).

Proving the inequality (ii) in (b) \(\varliminf_{n \rightarrow \infty} s_{n} t_{n} \geq s \bar{t}\)

For any fixed positive integer \(n\), since \(s_n \leq 0\) and \(t_n \geq 0\), we have: \(s_{n}t_{n} \geq \underline{s}\bar{t}\) because \(\underline{s}\) is the limit inferior of \((s_n)\), and \(\bar{t}\) is the limit superior of \((t_n)\). Now taking the limit inferior of both sides as \(n\rightarrow \infty\), we get: \(\varliminf_{n \rightarrow \infty} s_{n} t_{n} \geq \underline{s} \bar{t}\).

Key concepts

Limit Superior

The limit superior, often noted as \( \varlimsup \), is an important concept in real analysis that helps to describe the boundary behavior of a sequence. Imagine you have a sequence \((a_n)\) of real numbers. The limit superior \( \varlimsup_{n \rightarrow \infty} a_n \) is the largest amount or value that the terms of the sequence approach as you go further along the sequence.The steps involved in understanding the limit superior include:

• Outer Limits: Consider the greatest value that a subsequence can get infinitely close to. Think of it as the highest possible ceiling limit that cannot be surpassed by the sequence.
• Convergence: Occasionally, the sequence might approach a specific number with some variation. The limit superior will catch this highest approaching balance.

If a sequence \((a_n)\) approaches certain peaks, the limit superior identifies this peak value. It's often used in conjunction with sequences that do not have clear limits, helping students discern overall trend behaviors, just as it does with sequences that appear erratic or oscillating.

Limit Inferior

The limit inferior, denoted by \( \varliminf \), is another crucial idea in real analysis used in examining the behavior of sequences, similar to the limit superior, but in reverse. Here, it focuses on the smallest value that the sequence approaches infinitely often.Key points about limit inferior:

• Lower Bound: The limit inferior can be thought of as the lowest floor value that is infinitely close through some subsequence.
• Dynamic Behavior: When a sequence dips down to some lesser values, the limit inferior reflects this behavior where it approaches the lower trends.

Consider a sequence \((b_n)\) that appears to dip into certain troughs. The limit inferior \( \varliminf_{n \rightarrow \infty} b_n \) captures the smallest value along which these dips congregate. This concept is useful for sequences which don't neatly converge to a single point but instead highlight their minimum trend points.

Extended Reals

The concept of extended reals provides a broader scope to the traditional set of real numbers, \( \mathbb{R} \). The extended real number system includes two special elements that are not finite real numbers: positive infinity (\( +\infty \)) and negative infinity (\( -\infty \)). By including these elements, mathematicians can address situations involving limit processes more effectively.Important aspects of extended reals:

• Infinite Elements: The introduction of \( +\infty \) and \( -\infty \) helps in defining limits that might not otherwise exist in the usual real number sense, such as sequences that increase indefinitely.
• Handling Limits: It provides an elegant framework to handle sequences and series that diverge, integrating them into analyses succinctly.

Using the extended reals makes it possible to discuss limits and boundaries of sequences that do not simply converge to a standard real number, offering a robust terminology to discuss divergent behaviors or those reaching infinite gradients.