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The value is never more than 1, so the sequence is bounded above by 1. The terms are all positive, so the sequence is bounded below by 0.

Since the sequence has both upper and lower bounds, it's a bounded sequence.

Example 2

Is the sequence a_{n} = -n^{2} bounded or unbounded?

The terms are all negative, so the sequence is bounded above by 0. However, the terms get farther and farther from 0 with no lower bound. Since the sequence has no lower bound, it's unbounded.

Example 3

Is the sequence a_{n} = (-1)^{n} n bounded or unbounded?

The terms of this sequence get farther from 0, bouncing between positive and negative. This sequence has no upper or lower bound, so it's unbounded.

The words bounded and unbounded are opposites. A sequence either has both upper and lower bounds, or it doesn't.