Alternative Definitions for the Limit Sup/Inf of a Seq. of Real Numbers

# Alternative Definitions for the Limit Superior/Inferior of a Sequence of Real Numbers

Recall from The Limit Superior and Limit Inferior of a Sequence of Real Numbers page that if $(a_n)_{n=1}^{\infty}$ is a sequence of real numbers then we defined the limit superior of $(a_n)_{n=1}^{\infty}$ to be:

(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, we defined the limit inferior of $(a_n)_{n=1}^{\infty}$ to be:

(2)
\begin{align} \quad \liminf_{n \to \infty} a_n = \lim_{n \to \infty} \left ( \inf_{k \geq n} \{ a_k \} \right ) \end{align}

We will now look at some equivalent definitions that are often used to define the limit superior and limit inferior of a sequence of real numbers. Sometimes we may use:

(3)
\begin{align} \quad \limsup_{n \to \infty} a_n = \inf_{n \geq 1} \left \{ \sup_{k \geq n} \left \{ a_k \right \} \right \} \end{align}
(4)
\begin{align} \quad \liminf_{n \to \infty} a_n = \sup_{n \geq 1} \left \{ \inf_{k \geq n} \left \{ a_k \right \} \right \} \end{align}

Noticing that $\left ( \sup_{k \geq n} \left \{ a_k \right \} \right )_{n=1}^{\infty}$ is a decreasing sequence of real numbers, the above definition makes sense (since a decreasing sequence of real numbers converges to its infimum.