Table of Contents

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,