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 2dimensional 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:
The theorem is closely related to the theorem of HeineBorel.
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