Mathematical Analysis

Sequences and Series

What are the bounds of a sequence?

Once again, take the sequence {xn}n≥1. This sequence is bounded above if and only if there is a number M such that xnM (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 xn ≥ 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.


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