Let \((a_n)_{n=1}^\infty\) be a sequence.

1. We say that \((a_n)_{n=1}^\infty\) is strictly increasing if \(a_1 < a_2 < a_3 < ...\cdot\)
2. We say that \((a_n)_{n=1}^\infty\) is weakly increasing if \(a_1 \leq a_2 \leq a_3 \leq ...\cdot\)
3. We say that \((a_n)_{n=1}^\infty\) is strictly decreasing if \(a_1 > a_2 > a_3 > ...\cdot\)
4. We say that \((a_n)_{n=1}^\infty\) is weakly decreasing if \(a_1 \geq a_2 \geq a_3 \geq ...\cdot\)

If a sequence has any of these four properties, then we say that it is monotone

Suppose \(A\) is a set of numbers. We say that a number \(b\) is an upper bound for \(A\) if for every \(x \in A, x \leq b\). Similarly, b is a lower bound for \(A\) if for every \(x \in A, x \geq b\).

• Precise defintion of greatest lower bound and least upper bound

Suppose that \(A\) is a nonempty set of real numbers. That is, \(A\) is a set of numbers that has at least one element.

1. If \(A\) has an upper bound, then it has a least lower bound.
2. If \(A\) has an lower bound, then it has a greatest lower bound.

Suppose \((a_n)_{n=1}^\infty\) is a sequence, and let \(A\) be a set whose elements are the terms of the sequence.

1. If \((a_n)_{n=1}^\infty\) is weakly increasing and \(A\) has an upper bound, then \((a_n)_{n=1}^\infty\) converges to the least upper bound of \(A\).
2. If \((a_n)_{n=1}^\infty\) is weakly increasing and \(A\) does not have an upper bound, then \(lim_{n \to \infty} a_n = \infty\)
3. If \((a_n)_{n=1}^\infty\) is weakly decreasing and \(A\) has a lower bound, then \((a_n)_{n=1}^\infty\) converges to the greatest lower bound of \(A\).
4. If \((a_n)_{n=1}^\infty\) is weakly decreasing and \(A\) does not have a lower bound, then \(\lim_{n \to \infty} a_n = -\infty\)

Suppose \(u < v\), \(u\) is not an upper bound for \(A\), and \(v\) is an upper bound for \(A\). Then there are numbers \(u'\) and \(v'\) such that \(u \leq u' < v' \leq v\), \(v' - u' = (v-u)/2\), \(u'\) is not an upper bound for \(A\), and \(v'\) is an upper bound for \(A\).