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Determine Upper and Lower Bounds

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Five sequences are given. The problem is to determine if each has a lower bound and if each has an upper bound.

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Detailed explanations are given for determining if sequences have upper bounds and lower bounds.

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To determine whether or not a set has a least upper bound, we want to determine if there is a largest number in the set, or if there is a smallest number that we can get as close to as we like from below, but never reach (an upper limit). Likewise, the greatest lower bound is the smallest number in the set or it is a limit below, that is, it is a number that is smaller than any number in the set but that we can get as close to, from above, as we like.

We find limits of sequences the same way that we find limits of functions, but we need a little additional analysis here.

First we want to see if a set has a largest and smallest number.
In both (a) and (b), the numbers are getting smaller and 1 is the largest number in the set. So, 1 is the least upper bound.

To find the greatest lower bound in (a), ...

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