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Chapter 2: Problem 24

Suppose \(\left\\{x_{i}\right\\}_{i=m}^{\infty}\) diverges to \(-\infty .\) Show that every subsequence \(\left\\{x_{n_{k}}\right\\}_{k=1}^{\infty}\) of \(\left\\{x_{i}\right\\}_{i=m}^{\infty}\) also diverges to \(-\infty\).

Short Answer

Expert verified
Every subsequence \( \{x_{n_k}\}\) also diverges to \(-\infty\) just like the original sequence.

Step by step solution

01

Understand the Definition of Divergence

A sequence \( \{x_i\}_{i=m}^{\infty} \) diverges to \(-\infty \) if for every real number \( M \), there exists an index \( N \) such that for all \( i \geq N \), \( x_i < M \). This means that the terms of the sequence eventually become smaller than any chosen real number \( M \).
02

Identify the Subsequence

Consider any subsequence \( \{x_{n_k}\}_{k=1}^{\infty} \) of \( \{x_i\}_{i=m}^{\infty} \). By definition, for each \( k \), \( n_k \geq m \), ensuring it is drawn from the original sequence.
03

Apply Divergence to Subsequences

Since \( \{x_i\}_{i=m}^{\infty} \) diverges to \(-\infty \), for any real number \( M \), there exists an index \( N \) such that for all indices \( i \geq N \), \( x_i < M \). Since subsequences are formed by selecting indices from the original sequence, this property applies to subsequences as well.
04

Show That the Subsequence Diverges

For the subsequence \( \{x_{n_k}\}\), since \( n_k \geq m \) and \( n_k \to \infty \) as \( k \to \infty \), there will be a point \( k_0 \) such that for all \( k \geq k_0 \), \( n_k \geq N \). Consequently, \( x_{n_k} < M \), showing that \( \{x_{n_k}\}\) diverges to \(-\infty\).
05

General Conclusion

Since the property of diverging to \(-\infty\) is shown for any chosen \( M \) and any subsequence \( \{x_{n_k}\}\), it concludes that every subsequence of \( \{x_i\}_{i=m}^{\infty} \) must also diverge to \(-\infty\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Subsequence
A **subsequence** is a sequence that is derived from another sequence by deleting some or no elements without changing the order of the remaining elements. Think of it as picking some numbers from a series without rearranging them. For example, from the sequence of natural numbers \( \{1, 2, 3, 4, 5, \ldots\} \), a subsequence could be \( \{2, 4, 6, \ldots\} \), which is the sequence of even numbers.
  • A subsequence maintains the order of elements from the original sequence.
  • Each element in the subsequence comes from the original sequence.
  • Subsequences can "skip" elements but cannot "reorder" them.
When discussing the divergence of subsequences, especially in mathematical contexts, it is important to know that they inherit some properties from their parent sequences. This is crucial for proving how a subsequence diverges when the original sequence diverges.
Real Number
A **real number** is any number that can be found on the number line. This includes all the rational numbers, like fractions and integers, and all the irrational numbers, like \( \sqrt{2} \) or \( \pi \). Real numbers are used to express continuous quantities and are essential in understanding concepts in calculus and analysis.
  • Real numbers can be either positive, negative, or zero.
  • They encompass rational numbers, such as 3/4, and irrational numbers, such as \( \pi \).
  • In calculus, real numbers provide the values for limits and help in defining convergence and divergence.
In the context of sequences, when we say for every real number \( M \), it indicates a condition that applies across all possible values in the continuum of the real number system. This underlines the comprehensive nature of divergence to negative infinity.
Index
The **index** is essentially a counter or marker used to identify the position of items in a sequence. In mathematical sequences, we often use an index, denoted by integers, to specify which element we鈥檙e referring to.
  • In the sequence \( \{x_i\}_{i=m}^{\infty} \), \( i \) is the index.
  • The index helps in identifying the elements when discussing limits or convergence.
  • As \( i \) increases, it illustrates the progression through the sequence.
The role of the index becomes crucial when demonstrating a sequence's behavior at infinity. For sequences diverging to \(-\infty\), the concept of an index \( N \) where all \( i \geq N \) satisfies a condition, shows that from a certain position onwards, the sequence obeys a particular rule. This allows us to apply this condition to subsequences as well.
Negative Infinity
**Negative infinity** is a concept in calculus and analysis representing a value that is indefinitely small. It is not a real number but a way to describe behavior or trends as quantities decrease without bound.
  • In terms of sequences, a sequence diverging to negative infinity means its terms get smaller and smaller.
  • For every real number \( M \), the terms of the sequence eventually fall below \( M \).
  • It's used to describe the trend of a sequence where elements continue decreasing without limit.
When we say a sequence \( \{x_i\} \) diverges to \(-\infty\), it means that no matter how small the real number you choose (i.e., \( M \)), you can find an index in the sequence beyond which all subsequent terms are less than that chosen number. This characteristic remains true for any subsequence derived from the original sequence.

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Most popular questions from this chapter

Show that for any positive integer \(k\) $$ \lim _{n \rightarrow \infty} \frac{1}{n^{k}}=0 $$

Suppose the sequence \(\left\\{x_{i}\right\\}_{i=m}^{\infty}\) is not bounded. Show that either \(-\infty\) or \(+\infty\) is a subsequential limit of \(\left\\{x_{i}\right\\}_{i=m}^{\infty}\).

Suppose \(\sum_{i=m}^{\infty} a_{i}\) and \(\sum_{i=m}^{\infty} b_{i}\) are infinite series for which \(a_{i} \geq 0\) and \(b_{i}>0\) for all \(i \geq m .\) Show that if \(\sum_{i=m}^{\infty} a_{i}\) diverges and $$ \lim _{i \rightarrow \infty} \frac{a_{i}}{b_{i}}<+\infty $$ then \(\sum_{i=m}^{\infty} b_{i}\) diverges.

Show that \(\lim _{n \rightarrow \infty}\left|a_{n}\right|=0\) if and only if \(\lim _{n \rightarrow \infty} a_{n}=0\).

a. Show that \(\lim _{n \rightarrow \infty} \frac{1}{n}=0\). b. Show that $$ \lim _{n \rightarrow \infty} \frac{1}{n^{2}}=0 $$ by (i) using the definition of limit directly and then (ii) using previous results.
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