Limit Superior/Inferior of Sequences of Real Numbers Review

# Limit Superior/Inferior of Sequences of Real Numbers Review

We will now review some of the recent material regarding the limit superior and limit inferior of sequences of real numbers.

(1)
\begin{align} \quad \limsup_{n \to \infty} a_n = \lim_{n \to \infty} \left ( \sup_{k \geq n} \{ a_k \} \right ) \end{align}
• Similarly, the Limit Inferior of $(a_n)_{n=1}^{\infty}$ is defined as:
(2)
\begin{align} \quad \liminf_{n \to \infty} a_n = \lim_{n \to \infty} \left ( \inf_{k \geq n} \{ a_k \} \right ) \end{align}
(3)
\begin{align} \quad \limsup_{n \to \infty} a_n = \inf_{n \geq 1} \left \{ \sup_{ k\geq n} \{ a_k \} \right \} \end{align}
(4)
\begin{align} \quad \liminf_{n \to \infty} a_n = \sup_{n \geq 1} \left \{ \inf_{k \geq n} \{ a_k \} \right \} \end{align}
(5)
\begin{align} \quad \limsup_{n \to \infty} (-a_n) = -\liminf_{n \to \infty} a_n \end{align}
(6)
\begin{align} \quad \liminf_{n \to \infty} a_n \leq \limsup_{n \to \infty} a_n \end{align}
• Secondly, if $(a_n)_{n=1}^{\infty}$ and $(b_n)_{n=1}^{\infty}$ are both sequences such that $a_n \leq b_n$ for all $n \in \mathbb{N}$ (in fact, for sufficiently large $n$ works) then we have that:
(7)
\begin{align} \quad \liminf_{n \to \infty} a_n \leq \liminf_{n \to \infty} b_n \quad \mathrm{and} \quad \limsup_{n \to \infty} a_n \leq \limsup_{n \to \infty} b_n \end{align}
(8)
\begin{align} \quad \limsup_{n \to \infty} (a_n + b_n) \leq \limsup_{n \to \infty} a_n + \limsup_{n \to \infty} b_n \end{align}
(9)
\begin{align} \quad \liminf_{n \to \infty} (a_n + b_n) \geq \liminf_{n \to \infty} a_n + \liminf_{n \to \infty} b_n \end{align}
• Furthermore, if $(a_n)_{n=1}^{\infty}$ and $(b_n)_{n=1}^{\infty}$ are positive sequences of real numbers, then we can further conclude that:
(10)
\begin{align} \quad \limsup_{n \to \infty} (a_nb_n) \leq \left ( \limsup_{n \to \infty} a_n \right ) \left ( \limsup_{n \to \infty} b_n \right ) \end{align}
(11)
\begin{align} \quad \liminf_{n \to \infty} (a_nb_n) \geq \left ( \liminf_{n \to \infty} a_n \right ) \left ( \liminf_{n \to \infty} b_n \right ) \end{align}
(12)
\begin{align} \quad \liminf_{n \to \infty} \frac{a_{n+1}}{a_n} \leq \liminf_{n \to \infty} (a_n)^{1/n} \leq \limsup_{n \to \infty} (a_n)^{1/n} \leq \limsup_{n \to \infty} \frac{a_{n+1}}{a_n} \end{align}
(13)
\begin{align} \quad (a_n)_{n=1}^{\infty} \: \mathrm{converges \: to \:} A \in \mathbb{R} \quad \Leftrightarrow \quad \liminf_{n \to \infty} a_n = A = \limsup_{n \to \infty} a_n \end{align}
(14)
\begin{align} \quad (a_n)_{n=1}^{\infty} \: \mathrm{properly \: diverges \: to \:} \infty \quad \Leftrightarrow \quad \liminf_{n \to \infty} a_n = \infty = \limsup_{n \to \infty} a_n = \infty \end{align}
(15)
\begin{align} \quad (a_n)_{n=1}^{\infty} \: \mathrm{properly \: diverges \: to \:} -\infty \quad \Leftrightarrow \quad \liminf_{n \to \infty} a_n = -\infty = \limsup_{n \to \infty} a_n = \infty \end{align}