Question

Prove that (i) if \(\lim x_{n}=+\infty\) and \((\forall n) x_{n} \leq y_{n},\) then also \(\lim y_{n}=+\infty,\) and (ii) if \(\lim x_{n}=-\infty\) and \((\forall n) y_{n} \leq x_{n},\) then also \(\lim y_{n}=-\infty\).

Short answer

Both statements are true.

Step-by-step solution

Understand the Limit Theme

This exercise deals with sequences going to positive or negative infinity. A sequence \( \lim x_n = +\infty \) implies that for any arbitrary large number \( M > 0 \), there exists an integer \( N \) such that for all \( n > N \), \( x_n > M \). Similarly, \( \lim x_n = -\infty \) implies that for any arbitrarily small number \( M < 0 \), there exists an integer \( N \) such that for all \( n > N \), \( x_n < M \). This concept will be crucial in proving both statements.

Prove Statement (i) - Limit to Positive Infinity

We know \( \lim x_n = +\infty \). For any \( M > 0 \), there is an \( N \) such that if \( n > N \), then \( x_n > M \). Given \( x_n \leq y_n \) for all \( n \), it follows that \( y_n \geq x_n > M \). Thus, \( y_n > M \) for all \( n > N \). Therefore, \( \lim y_n = +\infty \) because \( y_n \) eventually becomes greater than any arbitrarily large positive number \( M \).

Prove Statement (ii) - Limit to Negative Infinity

We now begin with \( \lim x_n = -\infty \). For any \( M < 0 \), there is an \( N \) such that for all \( n > N \), \( x_n < M \). Given \( y_n \leq x_n \) for all \( n \), it follows that \( y_n \leq x_n < M \). Hence, \( y_n < M \) for all \( n > N \). Therefore, \( \lim y_n = -\infty \) because \( y_n \) eventually becomes less than any arbitrarily small negative number \( M \).

Key concepts

Sequence Convergence

In mathematics, understanding the notion of sequence convergence is fundamental. A sequence converges if its terms get closer and closer to a certain value as you progress through the sequence. One of the most common forms of convergence is when the terms of a sequence approach a finite limit that we call 'L', represented as \( \lim_{n \to \infty} x_n = L \). This means that for any tiny number \( \epsilon > 0 \), there exists an integer \( N \) such that for all \( n > N \), the terms of the sequence \( x_n \) are within \( \epsilon \) units of L. This is the formal definition of convergence.

However, in dealing with sequences that tend to positive or negative infinity, the sequence does not converge to a finite number. Instead, we say that a sequence converges to \( +\infty \) if, given any large positive number \( M \), there exists an \( N \) such that for all \( n > N \), \( x_n > M \). Similarly, a sequence converges to \( -\infty \) if there exists an \( N \) such that for all \( n > N \), \( x_n < M \), for any small negative number \( M \). This infinite behavior still represents a form of convergence in analysis, even though it does not have a finite limit.

Infinity in Sequences

Infinity in sequences unfolds as an intriguing concept, pivotal for understanding advanced mathematical analysis. Sequences that trend towards infinity don't really "reach" infinity, but they grow beyond any finite bound you can set. The statement \( \lim x_n = +\infty \) signifies that no matter how large a number you choose, the sequence willultimately surpass it as \( n \) progresses. Similarly, \( \lim x_n = -\infty \) means that the sequence will eventually dip below any small negative threshold.

This behavior is crucial when analyzing sequences, particularly when using infinitely increasing or decreasing sequences to establish inequalities or bounds within mathematical proofs or real-world applications. Understanding how sequences behave as they diverge toward infinity can help us determine the behavior of mathematical models over time.

• Allows for establishing boundaries for functions or sequences
• Helps understand divergence in financial models or scientific data
• Provides a way to formalize notions of growth rate applications

Inequalities in Sequences

Inequalities in sequences are a powerful mathematical tool for analyzing and comparing sequence behaviors, especially when limits are involved. They enable us to leverage existing information about a sequence to draw conclusions about related sequences.

For instance, if you know that \( x_n \leq y_n \) for all \( n \) and that \( \lim x_n = +\infty \), you can deduce that \( \lim y_n = +\infty \) as well. This is because as \( x_n \) grows without bound, so must \( y_n \), given it is always greater than or equal to \( x_n \). The same logic applies in reverse for sequences trending to \(-\infty\) where \( y_n \leq x_n \).

• This property shows how inequalities can be used to prove convergence behaviors.
• They are essential for understanding the stability or boundedness of complex systems.
• Inequalities provide a simple but crucial method for checking consistency in various mathematical contexts.

By understanding and applying inequalities, one can determine or estimate the behavior of sequences, especially in theoretical proofs and real-world problem settings.