Problem 14 - Basic Analysis: Introduction to Real Analysis

Question

Let \(\left\\{x_{n}\right\\}\) be a sequence. a) Show that \(\lim x_{n}=\infty\) if and only if \(\liminf x_{n}=\infty\). b) Then show that \(\lim x_{n}=-\infty\) if and only if limsup \(x_{n}=-\infty\). c) If \(\left\\{x_{n}\right\\}\) is monotone increasing, show that either \(\lim x_{n}\) exists and is finite or \(\lim x_{n}=\infty .\) In either case, \(\lim x_{n}=\sup \left\\{x_{n}: n \in \mathbb{N}\right\\}\)

Step-by-Step Solution

Answer

Prove \(\lim x_n = \infty\) if \(\liminf x_n = \infty\), \(\lim x_n = -\infty\) if \(\limsup x_n = -\infty\), and \(\lim x_n = \sup \{x_n\}\) for increasing sequences.

Step 1: Understanding Terms

Begin by understanding the definitions of \(\lim, \limsup, \liminf, \to \infty, \to -\infty\) for sequences. The limit of a sequence \(\{x_n\}\) approaches \(\infty\) if for every constant \(M\), there is an index \(N\) such that for all \(n > N\), \(x_n > M\). \(\limsup x_n\) is the greatest limit point, and \(\liminf x_n\) is the smallest limit point of the sequence \(\{x_n\}\).

Step 2a: Prove that \(\lim x_n = \infty \implies \liminf x_n = \infty\)

If \(\lim x_n = \infty\), then for every \(M\), there exists \(N\) such that for all \(n > N\), \(x_n > M\). This means that no terms are below any finite number for sufficiently large \(n\), hence \(\liminf x_n = \infty\).

Step 2b: Prove that \(\liminf x_n = \infty \implies \lim x_n = \infty\)

Assume \(\liminf x_n = \infty\). For every \(M\), there exists \(N\) such that for all \(n > N\), \(x_n > M\), confirming that \(\lim x_n = \infty\). Thus, both \(\lim x_n = \infty\) and \(\liminf x_n = \infty\) are equivalent.

Step 3a: Prove that \(\lim x_n = -\infty \implies \limsup x_n = -\infty\)

If \(\lim x_n = -\infty\), then for every constant \(M\), there exists \(N\) such that for all \(n > N\), \(x_n < M\). Therefore, \(\limsup x_n < M\) for all finite \(M\), which implies \(\limsup x_n = -\infty\).

Step 3b: Establish \(\limsup x_n = -\infty \implies \lim x_n = -\infty\)

If \(\limsup x_n = -\infty\), for every \(M\), there exists \(N\) such that for all \(n > N\), \(x_n < M\). This implies \(\lim x_n = -\infty\). Therefore, both \(\lim x_n = -\infty\) and \(\limsup x_n = -\infty\) are equivalent.

Step 4: Analyze Monotone Increasing Sequences

For a monotone increasing sequence \(\{x_n\}\), if the sequence is bounded above, it converges to a finite limit. Otherwise, it diverges to \(\infty\). Either case establishes \(\lim x_n = \sup \{x_n : n \in \mathbb{N}\}\). The supremum, without an upper bound, is \(\infty\).

Key Concepts

Monotone Increasing Sequences

A sequence is called **monotone increasing** if each term is at least as large as the term before it. Simply put, as you move forward in the sequence, the numbers never decrease. This property helps us make conclusive statements about its limit.

There are two possible behaviors for a monotone increasing sequence:

• If the sequence is bounded above, meaning there's a specific number that the sequence never exceeds, it must converge. This means as you add more and more terms, they get closer to some fixed, finite number.
• If it's not bounded above, the sequence will increase without end, diverging to infinity.

This is because, for any monotone increasing sequence, the limit, if it exists, will be the supremum (or the least upper bound) of the sequence. In simple terms, it's the smallest number that is greater than or equal to every term in the sequence.

Liminf and Limsup

The concepts of **liminf** and **limsup** help us capture the behavior of sequences, particularly when conveying their long-term behavior is tricky with a simple limit.

Here's a simplified explanation:

• **Liminf (Limit Inferior):** It's the greatest lower bound for the limits of subsequences. Practically, it tells us the smallest value that our sequence can "settle" around as it progresses.
• **Limsup (Limit Superior):** This is the smallest upper limit for the limits of subsequences. In essence, it pinpoints the largest value that our sequence can "approach" over time.

These two constructs are especially useful for sequences that do not have a conventional limit. They give insights into the "envelope" of a sequence's values as it progresses indefinitely. In our exploration, if \({\{x_n\}\}\) approaches \( \infty \), both its liminf and the sequence diverge to infinity. Conversely, if the sequence spirals towards \(-\infty\), its limsup, too, plunges to \(-\infty\).

Divergence to Infinity

When we say a sequence **diverges to infinity**, it grows larger and larger without bound. Imagine counting numbers; they keep getting bigger without ever stopping.

This is exactly what happens when a sequence's limit is infinity. Here's how you can visualize it:

• **Infinite Growth:** For any number you can think of, no matter how large, the sequence eventually surpasses it. So, if someone gives you a massive number \(M\), there will be a point in the sequence (let's call it \(N\)) where every term after \(N\) is larger than \(M\).
• **Consistent Increase:** No matter how big \(n\) becomes, the terms \(x_n\) grow without restriction. The sequence doesn't "settle" around any fixed number but keeps moving upward.

This concept is foundational in mathematics, especially when dealing with sequences and series, as it highlights that not all sequences have a "limit" in the traditional sense. Diverging to infinity is one such case where the sequence never stabilizes, always reaching for higher values as it progresses.