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As we saw in the last chapter, when we graph several terms of a sequence, certain behavior may appear. We may become convinced, for whatever reason, that the sequence is unbounded. Or, we may believe that the sequence is bounded and we may even notice the sequence moving towards a particular horizontal line. But how do we check that what we believe is happening really is happening? And what does it mean to “move towards” a line? In this chapter, we’ll make all this precise with what are called \(\varepsilon - N\) proofs.

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Convergence of Sequences of Real Numbers

  • 章节

  • 发表日期：

    2026年

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  • Ulrich Daepp,
  • Pamela Gorkin

摘要

As we saw in the last chapter, when we graph several terms of a sequence, certain behavior may appear. We may become convinced, for whatever reason, that the sequence is unbounded. Or, we may believe that the sequence is bounded and we may even notice the sequence moving towards a particular horizontal line. But how do we check that what we believe is happening really is happening? And what does it mean to “move towards” a line? In this chapter, we’ll make all this precise with what are called \(\varepsilon - N\) proofs.