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Math Help - Sandwich theorem generalization

  1. #1
    MHF Contributor Also sprach Zarathustra's Avatar
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    Sandwich theorem generalization

    Let {a_n}, {b_n} and {c_n} be a 3 closed sequences. Prove the generalization of sandwich rule:
    If a_n <= b_n <= c_n for all natural n and if liminf(a_n)=limsup(c_n)=L, then all the three sequences are convergent to L.


    I need the exact proof.
    Thank you very much!
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  2. #2
    MHF Contributor Bruno J.'s Avatar
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    Well

    L=\liminf a_n \leq \limsup a_n \leq \limsup c_n = L

    hence \limsup a_n = \liminf a_n = L, so \{a_n\} is convergent and \lim a_n = L.

    By a very similar argument you get that \{c_n\} converges to L. Now just apply the usual sandwich theorem. Hope this helps!
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