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Chapter 3: Problem 1

Show that if \(\left(x_{n}\right)\) is an unbounded sequence, then there exists a properly divergent subsequence.

Short Answer

Expert verified
The proof shows that for an unbounded sequence there always exists an increasing sequence of integers such that the subsequence, consisting of sequence elements at these integer positions, will properly diverge to infinity.

Step by step solution

01

Define Unbounded Sequence

An unbounded sequence is defined as a sequence where, for every real number \( M \) that can be chosen, there exists an element \( x_n \) such that \( |x_n| > M \). We say that such a sequence is unbounded.
02

Define Properly Divergent Sequence

A properly divergent sequence is a sequence that tends towards positive or negative infinity as \( n \rightarrow \infty \). This means that, for every real number \( M \), there will exist some number \( N \) such that if \( n > N \) then \( |x_n| > M \). It is clear that a properly divergent sequence is also unbounded.
03

Proof of Existence of Properly Divergent Subsequence

Assume you have an unbounded sequence. Then for every positive integer \( k \), there exists an integer \( n_k \) such that \( |x_{n_k}| > k \). Among these, choose \( n_k \) so that it is greater than any previously chosen \( n_i \) for \( i < k \). This gives us an increasing sequence of integers \( (n_k) \). Subsequence \( |x_{n_k}| \) will now be greater than \( k \) for all \( k \) which means it properly diverges to infinity, thereby proving the given statement.

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Most popular questions from this chapter

Let \(A\) be an infinite subset of \(\mathbb{R}\) that is bounded above and let \(u:=\) sup \(A\). Show there exists an increasing sequence \(\left(x_{n}\right)\) with \(x_{n} \in A\) for all \(n \in \mathbb{N}\) such that \(u=\lim \left(x_{n}\right)\).

Prove that if \(\lim \left(x_{n}\right)=x\) and if \(x>0\), then there exists a natural number \(M\) such that \(x_{n}>0\) for all \(n \geq M\).

Investigate the convergence or the divergence of the following sequences: (a) \(\left(\sqrt{n^{2}+2}\right)\). (b) \(\left(\sqrt{n} /\left(n^{2}+1\right)\right)\). (c) \(\left(\sqrt{n^{2}+1} / \sqrt{n}\right)\), (d) \((\sin \sqrt{n})\).

Let \(a>0\) and let \(z_{1}>0\). Define \(z_{n+1}:=\sqrt{a+z_{n}}\) for \(n \in \mathbb{N}\). Show that \(\left(z_{n}\right)\) converges and find the limit.

Establish the convergence or the divergence of the sequence \(\left(y_{n}\right)\), where $$y_{n}:=\frac{1}{n+1}+\frac{1}{n+2}+\cdots+\frac{1}{2 n} \quad \text { for } \quad n \in \mathbb{N}$$
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