A bounded sequence is one where the absolute value of every term is less than or equal to a particular real, positive number. You can think of it as there being a well defined boundary line such that no term in the sequence can be found on the outskirts of that line.
More formally, a sequence X is bounded if there is a real number, M greater than 0, such |xn| ≤ M for all n ε N.
Examples of Bounded Sequences
One example of a sequence that is bounded is the one defined by”
The right hand side of this equation tells us that n is indexed between 1 and infinity. This makes the sequence into a sequence of fractions, with the numerators always being one and the denominators always being numbers that are greater than one. A basic algebraic identity tells us that x-k = 1/xk. So each term in the sequence is a fractional part of one, and we can say that for every term in our sequence, |xn| ≤ 1.
Remember now our definition of a bounded sequence: a sequence X is bounded if there is a real number, M greater than 0, such |xn| ≤ M for all n ε N. Let M = 1, and then M is be a real number greater than zero such that |xn| ≤ M for all n between 1 and infinity. So our sequence is bounded.
Bounded Sequences and Convergence
Every absolutely convergent sequence is bounded, so if we know that a sequence is convergent, we know immediately that it is bounded. Note that this doesn’t tell us anything about whether a bounded sequence is convergent: it may or may not be. As an example, the sequence drawn above is not convergent though it is bounded.
Bounded Above and Below
If we say a sequence is bounded, it is bounded above and below. Some sequences, however, are only bounded from one side.
If all the terms of a sequence are less than or equal to a number K’ the sequence is said to be bounded above, and K’ is the upper bound. The least possible K is the supremum.
Gallup, Nathaniel. Mat25 Lecture 9 Notes: Boundedness of Sequences. Retrieved from https://www.math.ucdavis.edu/~npgallup/m17_mat25/lecture_notes/lecture_9/m17_mat25_lecture_9_notes.pdf on January 25, 2018.
Math Learning Center: Sequences. Retrieved from https://www3.ul.ie/cemtl/pdf%20files/cm2/BoundedSequence.pdf on January 26, 2018
Larson & Edwards. Calculus.