Let (an) be a sequence in ℝ. Define the limit superior (or the upper limit) of (an) by lim supn

step 1 The question was given the limit of the AN equal to 1 and the limit of the PN equal to minus 1.

Let (an) be a sequence in ℝ. Define the limit superior (or...

Let $\left(a_{n}\right)$ be a sequence in $\mathbb{R}$. Define the limit superior (or the upper limit) of $\left(a_{n}\right)$ by $$ \limsup _{n \rightarrow \infty} a_{n}:=\left\{\begin{array}{ll} \lim _{n \rightarrow \infty} M_{n} & \text { if }\left(a_{n}\right) \text { is bounded above and } a_{n} \not \rightarrow-\infty \\ \infty & \text { if }\left(a_{n}\right) \text { is not bounded above } \\ -\infty & \text { if } a_{n} \rightarrow-\infty \end{array}\right. $$ where the sequence $\left(M_{n}\right)$ is as defined in Exercise 35 . Similarly, define the limit inferior (or the lower limit) of $\left(a_{n}\right)$ by $$ \liminf _{n \rightarrow \infty} a_{n}:=\left\{\begin{array}{ll} \lim _{n \rightarrow \infty} m_{n} & \text { if }\left(a_{n}\right) \text { is bounded below and } a_{n} \rightarrow \infty \\ -\infty & \text { if }\left(a_{n}\right) \text { is not bounded below, } \\ \infty & \text { if } a_{n} \rightarrow \infty \end{array}\right. $$ where the sequence $\left(m_{n}\right)$ is as defined in Exercise 35 . If $\left(a_{n}\right)$ is bounded, then show that the set $C$ of all cluster points of $\left(a_{n}\right)$ is nonempty, and moreover, $$ \limsup _{n \rightarrow \infty} a_{n}=\lim _{n \rightarrow \infty} \sup \left\{a_{n}, a_{n+1}, \ldots\right\}=\max C $$ and $$ \liminf _{n \rightarrow \infty} a_{n}=\lim _{n \rightarrow \infty} \inf \left\{a_{n}, a_{n+1}, \ldots\right\}=\min C. $$

Let $\left(a_{n}\right)$ be a sequence in $\mathbb{R}$. Define the limit superior (or the upper limit) of $\left(a_{n}\right)$ by
$$
\limsup _{n \rightarrow \infty} a_{n}:=\left\{\begin{array}{ll}
\lim _{n \rightarrow \infty} M_{n} & \text { if }\left(a_{n}\right) \text { is bounded above and } a_{n} \not \rightarrow-\infty \\
\infty & \text { if }\left(a_{n}\right) \text { is not bounded above } \\
-\infty & \text { if } a_{n} \rightarrow-\infty
\end{array}\right.
$$
where the sequence $\left(M_{n}\right)$ is as defined in Exercise 35 . Similarly, define the limit inferior (or the lower limit) of $\left(a_{n}\right)$ by
$$
\liminf _{n \rightarrow \infty} a_{n}:=\left\{\begin{array}{ll}
\lim _{n \rightarrow \infty} m_{n} & \text { if }\left(a_{n}\right) \text { is bounded below and } a_{n} \rightarrow \infty \\
-\infty & \text { if }\left(a_{n}\right) \text { is not bounded below, } \\
\infty & \text { if } a_{n} \rightarrow \infty
\end{array}\right.
$$
where the sequence $\left(m_{n}\right)$ is as defined in Exercise 35 . If $\left(a_{n}\right)$ is bounded, then show that the set $C$ of all cluster points of $\left(a_{n}\right)$ is nonempty, and moreover,
$$
\limsup _{n \rightarrow \infty} a_{n}=\lim _{n \rightarrow \infty} \sup \left\{a_{n}, a_{n+1}, \ldots\right\}=\max C
$$
and
$$
\liminf _{n \rightarrow \infty} a_{n}=\lim _{n \rightarrow \infty} \inf \left\{a_{n}, a_{n+1}, \ldots\right\}=\min C.
$$

Key Concepts

Supremum and Infimum

The supremum (least upper bound) and infimum (greatest lower bound) of a set are fundamental concepts in real analysis. In the context of sequences, the supremum and infimum of the tail sets determine the limits that the sequence can approach from above and below. The limit superior is the limit of these supremums of the tails, while the limit inferior is the limit of the infimums, thereby directly linking these concepts to cluster points.

Cluster Points (Accumulation Points)

Cluster points of a sequence are those values that can be approached by some subsequence. In the context of a bounded sequence, these points exist due to the Bolzano-Weierstrass theorem. The set of all cluster points encapsulates the possible limit behaviors of the sequence, and notably, the maximum and minimum of this set coincide with the limsup and liminf, respectively.

Bounded Sequences

A bounded sequence is one whose terms are confined within some fixed interval, meaning there exist real numbers that serve as universal lower and upper bounds for all terms. Boundedness ensures, by results like the Bolzano-Weierstrass theorem, that the sequence has at least one cluster point, and it is crucial for the definitions of limsup and liminf in this context.

Limit Superior

The limit superior (limsup) of a sequence is defined as the eventual supremum of the tail sequences. It represents the greatest limit point (or accumulation point) that the sequence can approach, reflecting the upper bound of the fluctuating behavior of the sequence as it progresses indefinitely.

Limit Inferior

The limit inferior (liminf) of a sequence is the corresponding eventual infimum of the tail sequences. It captures the smallest accumulation point of the sequence, representing the lower bound of the oscillations or convergent behavior of the sequence as n approaches infinity.

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