Proving that $\liminf_{n\to\infty} x_n = \sup\left\{z : \{n : x_n < z\} \text{ is finite}\right\}$; what does the answer mean?

Question

Suppose $x_n$ is a bounded sequence.

Prove that $$\liminf_{n\to\infty} x_n = \sup\left\{z : \{n : x_n < z\} \text{ is finite}\right\}$$ I'm having trouble even interpreting what this is suppose to mean. Please could somebody explain it for me?

Answer 1

Maybe a proof of this can help you understand what this is supposed to mean.

Equivalent are:

• $y<\sup\left\{ z\mid\left\{ n\mid x_{n}<z\right\} \text{ is finite}\right\} $

• $\exists z\left[y<z\wedge\left\{ n\mid x_{n}<z\right\} \text{ is finite}\right]$

• $\exists z\left[y<z\wedge\exists n\; z\leq\inf_{k\geq n}x_{k}\right]$

• $\exists n\; y<\inf_{k\geq n}x_{k}$

• $y<\lim_{n\rightarrow\infty}\inf_{k\geq n}x_{k}=\liminf_{n\rightarrow\infty}x_{n}$

If the statements $y<a$ and $y<b$ are equivalent then $a=b$