Let (an) be a sequence in ℝ. (i) Assume that (an) is bounded above and an ↛-∞. Define Mn:=sup{an

step 1 higher here we have to prove that the sequence a in the real numbers has no conversion subsequen

Let (an) be a sequence in ℝ. (i) Assume that (an) is bounded...

Let $\left(a_{n}\right)$ be a sequence in $\mathbb{R}$. (i) Assume that $\left(a_{n}\right)$ is bounded above and $a_{n} \not \rightarrow-\infty$. Define $M_{n}:=\sup \left\{a_{n}, a_{n+1}, \ldots\right\}$ for $n \in \mathbb{N} \quad$ and $\quad M:=\inf \left\{M_{1}, M_{2}, \ldots\right\}$ Show that the sequence $\left(M_{n}\right)$ converges to $M$ and $M$ is the largest cluster point of $\left(a_{n}\right)$. (ii) Assume that $\left(a_{n}\right)$ is bounded below and $a_{n} \not \leftrightarrow \infty$. Define $m_{n}:=\inf \left\{a_{n}, a_{n+1}, \ldots\right\}$ for $n \in \mathbb{N} \quad$ and $\quad m:=\sup \left\{m_{1}, m_{2}, \ldots\right\}$ Show that the sequence $\left(m_{n}\right)$ converges to $m$ and $m$ is the smallest cluster point of $\left(a_{n}\right)$. [See Exercise 16 for the definition of a cluster point.]

Let $\left(a_{n}\right)$ be a sequence in $\mathbb{R}$.
(i) Assume that $\left(a_{n}\right)$ is bounded above and $a_{n} \not \rightarrow-\infty$. Define
$M_{n}:=\sup \left\{a_{n}, a_{n+1}, \ldots\right\}$ for $n \in \mathbb{N} \quad$ and $\quad M:=\inf \left\{M_{1}, M_{2}, \ldots\right\}$
Show that the sequence $\left(M_{n}\right)$ converges to $M$ and $M$ is the largest cluster point of $\left(a_{n}\right)$.
(ii) Assume that $\left(a_{n}\right)$ is bounded below and $a_{n} \not \leftrightarrow \infty$. Define
$m_{n}:=\inf \left\{a_{n}, a_{n+1}, \ldots\right\}$ for $n \in \mathbb{N} \quad$ and $\quad m:=\sup \left\{m_{1}, m_{2}, \ldots\right\}$
Show that the sequence $\left(m_{n}\right)$ converges to $m$ and $m$ is the smallest cluster point of $\left(a_{n}\right)$. [See Exercise 16 for the definition of a cluster point.]

Key Concepts

Convergence of Sequences

Convergence of a sequence means that as the index increases, the sequence's terms approach a single specific value. This concept is central to real analysis, as it allows one to relate the behavior of an entire sequence to its limits. The process of taking limits of the tail sequences (of suprema or infima) and linking these limits to the cluster points illustrates how local bounded behaviors can lead to global convergence properties.

Cluster Points

A cluster point (or accumulation point) of a sequence is a value such that every neighborhood around it contains infinitely many elements of the sequence. In the context of these tail sequences, the limit of the sequence of tail suprema represents the largest cluster point, while the limit of the sequence of tail infima represents the smallest cluster point. This connection is fundamental in the analysis of the asymptotic behavior of real sequences.

Monotonicity of Tail Sequences

Tail sequences defined by taking suprema or infima over all subsequent terms of the sequence yield monotonic sequences; for instance, the sequence of tail suprema is non?increasing and the sequence of tail infima is non?decreasing. This monotonicity, in combination with boundedness, guarantees convergence according to the Monotone Convergence Theorem, which is pivotal in proving that these sequences converge to their respective limits.

Supremum and Infimum

The supremum of a subset of the real numbers is the smallest number that is greater than or equal to every element of the set, while the infimum is the greatest number that is less than or equal to every element. These concepts are essential in understanding limits of sequences, particularly when considering the limits of tail suprema or tail infima which lead to least upper bounds or greatest lower bounds related to cluster points.

Bounded Sequences

A sequence is said to be bounded (either above or below) if all its elements lie within a fixed real number barrier. In analysis, having a bounded sequence ensures that the supremum (least upper bound) or the infimum (greatest lower bound) exists, which is a key prerequisite for defining limits via tail sequences and proving convergence properties.

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