What exactly is the meaning of the following $\inf\{ s_n : n > N\}$ and $\sup\{ s_n : n > N\}$

Question

What exactly is the meaning of the following $$u_N = \inf\{ s_n : n > N\} \ \ \text{ and} \ \ v_N = \sup\{ s_n : n > N\}$$

This might seem a stupid question, but I am not understanding the meaning of these sets. We know that the infimum and the supremum should be just one, so what is the meaning of the sets $u_N$ and $v_N$?

This all came out studying monotone sequences...

Answer 1

If $u_N$ and $v_N$ are finite, it mean that $$\forall \varepsilon>0, \exists n>N: u_N<s_n<u_N+\varepsilon$$ and $$\forall \varepsilon>0,\exists n>N: v_N-\varepsilon<s_n<v_N.$$

If for example $u_N$ is not finite, (and is for example $+\infty $), it mean that $$\forall M\in \mathbb N, \exists n>N: M< s_n.$$

Answer 2

The definition of limit superior and limit inferior you may find here, and an interesting post here.

As an example to elucidate you could take the sequence $x_n = 1 - \frac{1}{n}$. Defining $u_N$ and $v_N$ we have

$$u_N = \inf\,\,\{ 1 - \frac{1}{n}; n \geq N\} = \inf\,\,\{1 - \frac{1}{N}, 1 - \frac{1}{N+1}, 1- \frac{1}{N+2}, \ldots\} = 1 - \frac{1}{N}$$

and

$$v_N = \sup \,\,\{1 - \frac{1}{n} ; n \geq N\} = \sup\,\,\{1 - \frac{1}{N}, 1 - \frac{1}{N+1}, 1 - \frac{1}{N+2}, \ldots\} = 1$$

then $$\lim \inf x_n = \lim_{N \to \infty} u_N = 1 \ \ \text{and} \ \ \lim \sup x_n = \lim_{N \to \infty} v_N = 1$$