The Image Of A Sequence Is A Set

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Definition (The Image Of A Sequence Is A Set)

Given a sequence $(a_n)$ of real numbers, note that the image of the sequence, namely $a(\mathbb{N}) = \{a_n\mid n\in\mathbb{N}\}$, is a subset of the real numbers. Because the image of the sequence is a set of real numbers, we can use any of our previous words that we defined on sets of real numbers, and now apply them to a sequence. Here are some examples:

• We say a sequence is bounded if the image of the sequence is a bounded set.
• A lower bound for a sequence is a lower bound for the image of the sequence.
• The supremum of a sequence is the supremum of the image of the sequence.
• A limit point of a sequence is a limit point of the image of the sequence.