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Reciprocal of lim inf

  1. Feb 22, 2006 #1
    Suppose a_n is a bounded sequence. Then prove that lim sup a_n = 1/lim inf (1/a_n).

    This seems completely obvious to me, I don't know how to do this any simpler.
  2. jcsd
  3. Feb 22, 2006 #2


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    lim sup{an}
    = limn->oosupk>n{ak} ... (justify this)
    = limn->oo[1/infk>n{1/ak}] ... (justify this)
    = 1/[limn->ooinfk>n{1/ak}] ... (justify this)
    = 1/lim inf{1/an} ... (justify this)
    Last edited: Feb 22, 2006
  4. Feb 23, 2006 #3
    Let S={limit points of a_n}. Since a_n is strictly positive and bounded, the limit points of 1/a_n are precisely 1/s for s in S. It follows from there, doesn't it?
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