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The algebra of limits is a set of rules for how limits may be manipulated with other operators.

For real sequences

For real convergent sequences and where and , and for a real number :




If also and


Proof

  1. Fix
    Because for and
    because for
    Set
    By the triangle inequality,
    and
    So ie
  1. If , then , so it converges to 0. If , fix
    Because for
    So ie
  1. Fix As is convergent, it is bounded by some ie
    Because for and
    because for
    Set
    By the triangle inequality, and boundedness of , And
    So ie
  1. By part 3,
    Fix
    Because for
    Also, for
    By the triangle inequality,
    And for so
    Now
    And now
    Setting we have
    or in other words
    Therefore,
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