Page 160 - 'Blast_Into_Math

Blast into Math! Analatic nummer theora: ants, ghosts and giants

The relationship between boundedness for sets and boundedness for sequences is explained by the
following proposition.

Proposition 7.1.6 (Bounds). Let

∞
{x n } n=1

be a sequence of numbers. Then, the sequence is bounded above if and only if the set of elements in the
sequence is bounded above, and the sequence is bounded below if and only if the set of elements in the
sequence is bounded below. The sequence is bounded if and only if the set of elements in the sequence is
bounded.

Prove this proposition as an exercise. Hint: You will only need to use the definitions.

♥

An increasing sequence which is bounded is like a trail of mathematical ants on the number line who

are marching to the right, but may not cross an upper bound. Do they converge to a limit?

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