Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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


    User Avatar
    Science Advisor
    Homework Helper

    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?
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook