## Mathematical Analysis## Sequences and Series |

## What are the bounds of a sequence? |

Once again, take the sequence {*x*_{n}}_{n≥1}. This sequence is *bounded above* if and only if there is a number *M* such that *x*_{n} ≤ *M* (the *M* is called an *upper-bound).* In addition, the sequence is *bounded below* if and only if there is a number *m* such that *x _{n} ≥ m* (the

*m*is called a

*lower-bound).*For example, the sequence {2

^{n}}

_{n≥1}, is bounded below by 0 because it is positive, but not bounded above.

The sequence is usually said to be merely *bounded* (or “bd” for short) if both of the properties (upper- and lower-bound) hold. For example, the harmonic sequence {1, ½, 1/3, 1/4 …} is considered bounded because no term is greater than 1 or less than 0; thus, the upper- and lower-bounds, respectively, apply.