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 {2ⁿ}_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.