A real number a is called a cluster point of a sequence (an) in ℝ if there is a subsequence (ank

step 1 Here in this question, a sequence is given to us which is AK where K runs from 1 to infinity. An

A real number a is called a cluster point of a sequence (an)...

A real number $a$ is called a cluster point of a sequence $\left(a_{n}\right)$ in $\mathbb{R}$ if there is a subsequence $\left(a_{n_{k}}\right)$ of $\left(a_{n}\right)$ such that $a_{n_{k}} \rightarrow a$ (i) Show that if $a_{n} \rightarrow a$, then $a$ is the only cluster point of $\left(a_{n}\right)$. (ii) Show that the converse of (i) is not true. In other words, show that there is a divergent sequence that has a unique cluster point. (Hint: $a_{2 k}:=\frac{1}{2 k}$ and $a_{2 k+1}:=2 k+1$ for $k \in \mathbb{N}$.) (iii) Show that if $a_{n} \rightarrow \infty$ or if $a_{n} \rightarrow-\infty$, then $\left(a_{n}\right)$ has no cluster point. (iv) Show that the converse of (iii) is not true. In other words, show that there is a sequence without a cluster point that neither tends to $\infty$ nor tends to $-\infty .$ (Hint: $a_{n}:=(-1)^{n} n$ for $n \in \mathbb{N}$.)

A real number $a$ is called a cluster point of a sequence $\left(a_{n}\right)$ in $\mathbb{R}$ if there is a subsequence $\left(a_{n_{k}}\right)$ of $\left(a_{n}\right)$ such that $a_{n_{k}} \rightarrow a$
(i) Show that if $a_{n} \rightarrow a$, then $a$ is the only cluster point of $\left(a_{n}\right)$.
(ii) Show that the converse of (i) is not true. In other words, show that there is a divergent sequence that has a unique cluster point. (Hint:
$a_{2 k}:=\frac{1}{2 k}$ and $a_{2 k+1}:=2 k+1$ for $k \in \mathbb{N}$.)
(iii) Show that if $a_{n} \rightarrow \infty$ or if $a_{n} \rightarrow-\infty$, then $\left(a_{n}\right)$ has no cluster point.
(iv) Show that the converse of (iii) is not true. In other words, show that there is a sequence without a cluster point that neither tends to $\infty$ nor tends to $-\infty .$ (Hint: $a_{n}:=(-1)^{n} n$ for $n \in \mathbb{N}$.)

Key Concepts

Sequence Convergence

This concept deals with sequences whose terms approach a single limit value as the index goes to infinity. In analysis, if a sequence converges, every subsequence also converges to the same limit, which underpins the idea that the limit is unique. Convergence is foundational for understanding many other properties of sequences.

Subsequences

Subsequences are sequences derived by selecting certain elements from a given sequence without changing their order. They are crucial in defining cluster points, as a real number is considered a cluster point of a sequence if there exists a subsequence that converges to that number. This allows the study of accumulation behavior within sequences.

Cluster Points

A cluster point (or accumulation point) of a sequence is any real number that is the limit of some subsequence of the original sequence. This concept captures the idea of where the sequence 'clusters' or accumulates, even if the sequence itself does not converge. It is a key tool in understanding the limiting behavior of sequences that might otherwise seem erratic or divergent.

Divergence and Infinite Limits

Divergence refers to sequences that do not settle to a finite limit; they may diverge by oscillation or by tending to infinity (or minus infinity). Infinite limits are specific cases where the sequence grows without bound but, importantly, such behavior does not yield cluster points in the real numbers. Distinguishing between divergent behavior and the existence of cluster points deepens our understanding of sequence limits.

Counterexamples in Analysis

Counterexamples are specific instances constructed to show that a particular property or converse does not hold in general. In the context of sequences and cluster points, counterexamples demonstrate that while convergence implies a unique cluster point, the reverse is not necessarily true, and that sequences without any cluster points may not necessarily tend to infinity. These constructions are essential for highlighting the nuances and limits of theoretical results.

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