Monotone Sequences
The Monotone Sequence Theorem is one of only two major mechanisms by which it's possible to prove that a sequence converges without having to explicitly know what the value of the limit is.
Theorem (Monotone Sequence Theorem)
Suppose that \(\{a_n\}_{n=1}^\infty\) is a nondecreasing sequence which is bounded above. Then
\[ \lim_{n \rightarrow \infty} a_n = \sup \{ a_n \ : \ n \in {\mathbb N} \}, \]
i.e., the limit exists and is equal to the supremum of the set of terms. Similarly, if \(\{b_n\}_{n=1}^\infty\) is a nonincreasing sequence which is bounded below, then
\[ \lim_{n \rightarrow \infty} b_n = \sup \{ b_n \ : \ n \in {\mathbb N} \}. \]
Proof
1. Example: Defining \(k\)-th roots of positive real numbers
Theorem
For each nonnegative real number \(A\) and every positive integer \(k\), there is a real number \(x\) such that \(x^k = A\).
Proof
2. Limsup and Liminf
Definition
Given any sequence \(\{a_n\}_{n=1}^\infty\), we can define its limit superior and limit inferior by
\[ \limsup_{n \rightarrow \infty} a_n := \inf_{N \geq 1} \sup_{n \geq N} a_n \]
and
\[ \liminf_{n \rightarrow \infty} a_n := \sup_{N \geq 1} \inf_{n \geq N} a_n. \]
The limsup and liminf are always defined as extended real numbers. If the sequence \(a_n\) is not bounded above, we interpret \(\sup_{n \geq N} a_n\) as \(+\infty\) and say that \(\limsup_{n \rightarrow \infty} a_n = \infty\). Likewise, if the sequence is not bounded below, then we understand \(\liminf_{n \rightarrow \infty} a_n = -\infty\).
Theorem
For any sequence \(\{a_n\}_{n=1}^\infty\),
\[ \liminf_{n \rightarrow \infty} a_n \leq \limsup_{n \rightarrow \infty} a_n. \]
The limsup and liminf are equal and finite if and only if the sequence converges; in this case, the limsup and liminf simply equal the limit. Both limsup and liminf are \(+\infty\) if and only if the sequence diverges to \(+\infty\), and both are equal to \(-\infty\) iff the sequence diverges to \(-\infty\).
Proof
Exercises
- Let \(a_1 := 0\) and for each \(n > 1\), let \(a_n := \sqrt{3 + 2a_{n-1}}\). Prove that this sequence is monotone increasing and converges.
- Let \(b_n := \frac{1}{n^2} + (-1)^n \frac{1+n}{n}\). Compute \(\limsup_{n \rightarrow \infty} b_n\) and \(\liminf_{n \rightarrow \infty} b_n\) (and prove your answer).