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Lim inf an = -lim sup -an

  1. Oct 6, 2011 #1
    Let [itex]\{a_n\}[/itex] be a sequence of real numbers.

    Then [itex]\liminf\limits_{n \rightarrow \infty}[/itex] [itex] a_n = \limsup\limits_{n \rightarrow \infty}[/itex] [itex] -a_n [/itex]

    So my first strategy was to translate these into the definitions given in Rudin:

    [itex]\limsup\limits_{n \rightarrow \infty}[/itex] [itex]a_n = \sup_{n \geq 1}(\inf_{k \geq n} a_k)[/itex]

    From a homework problem in the first chapter, we know that [itex]\inf[/itex][itex]A=-\sup[/itex] [itex]-A[/itex]

    So applying that reasoning, I get:

    [itex]\sup_{n \geq 1}(-\sup_{k \geq n} -a_k)[/itex]

    But now I'm not sure how to mess with that sup in the front.
     
  2. jcsd
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