The sequence *a*_{1}, *a*_{2}, *a*_{3}, ... is called *bounded* if there exists a number *L* such that the absolute value |*a*_{n}| is less than *L* for every index *n*. Graphically, this can be imagined as points *a*_{i} plotted on a 2-dimensional graph, with *i* on the horizontal axis and the value on the vertical. The sequence then travels to the right as it progresses, and it is bounded if we can draw a horizontal strip which encloses all of the points.

A *subsequence* is a sequence which omits some members, for instance *a*_{2}, *a*_{5}, *a*_{13}, ...

Here is a sketch of the proof:

- start with a finite interval which contains all the
*a*_{n}. Since the sequence is bounded, the interval ( -L, L ) which we have from the definition will do. - Cut it into two halves. At least one half must contain
*a*_{n}for infinitely many*n*. - Then continue with that half and cut it into two halves, etc.
- This process constructs a sequence of intervals whose common element is limit of a subsequence.