[Math] What’s the relation between $\limsup$ and $\inf\sup$

limsup-and-liminf

I've found in my book that:
$$\liminf_{n\to\infty} \ x_{n} = \sup\{\inf\{x_{k}:k\geq n \}:n \in \mathbb{N}\}$$
$$\limsup_{n\to\infty} \ x_{n} = \inf\{\sup\{x_{k}:k\geq n \}:n \in \mathbb{N}\}$$

But I don't understand why. According to my book the definition of $\limsup$ and $\liminf$ is the following:

$$\limsup_{n\to\infty} \ s_{n} = \lim_{N\to\infty} \sup \{s_{n}:n > N \} $$

$$\liminf_{n\to\infty} \ s_{n} = \lim_{N\to\infty} \inf \{s_{n}:n > N \} $$

Best Answer

Let $X_n=\{x_{k}:k\geq n \}$. Then $X_{n+1} \subseteq X_n$ and therefore: $$ \inf X_n \le \inf X_{n+1} \le \sup X_{n+1} \le \sup X_n $$ This implies:

  • $a_n=\sup X_n$ is a decreasing sequence and so $\lim_n a_n = \inf_n a_n$.

  • $b_n=\inf X_n$ is an increasing sequence and so $\lim_n b_n = \sup_n b_n$.

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