## Examining Properties of a Bounded Sequence

We first examine some examples regarding the liminf and limsup of a bounded sequence, and we will discover theorems regarding them after.

Def: A bounded sequence $$(a_n)$$ has the property that for all $$n$$, we can define the following:

• $$I_N := \inf \{a_n, n > N \}$$
• $$S_N := \sup \{a_n, n > N\}$$

### Example 1

Take the example of $$a_n = (1, -1, 1/2, -1/2, ...)$$

$$I_N$$ $$S_N$$
$$I_1 = \inf\{ a_n, n>1 \} = -1$$ $$S_1 = \sup\{ a_n, n>1 \} = 1/2$$
$$I_2 = \inf\{ a_n, n>2 \} = -1/2$$ $$S_2 = \sup\{ a_n, n>2 \} = 1/2$$
$$I_3 = \inf\{ a_n, n>3 \} = -1/2$$ $$S_3 = \sup\{ a_n, n>3 \} = 1/3$$
$$I_4 = \inf\{ a_n, n>4 \} = -1/3$$ $$S_4 = \sup\{ a_n, n>4 \} = 1/3$$
$$...$$ $$...$$
$$I_n = (-1, -1/2, -1/2, -1/3...)$$ $$S_n = (1/2, 1/2, 1/3, 1/3, ...)$$
$$lim_{n\to\infty} I_N = 0$$ $$lim_{n\to\infty} S_N = 0$$

In this case, we observe that $$I_1 \leq I_2 \leq ... \leq \lim_{n\to\infty}I_n \leq \lim_{n\to\infty}a_n \leq \lim_{n\to\infty} S_n \leq ... \leq S_2 \leq S_1$$, so we can see that $$a_n$$ seems to converge.

### Example 2

Take the example of $$a_n = (-1, 1, -1, 1, ...)$$

Sequence Limit
$$I_N = -1\ \forall N$$ $$\lim_{n\to\infty} I_N = -1$$
$$S_N = 1\ \forall N$$ $$\lim_{n\to\infty} S_N = 1$$

In this case, we can see that $$a_n$$ seems to diverge.

## Lemma: Boundedness and Infimums and Supremums

Based on our observations above,