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Math Help - liminf of a product

  1. #1
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    liminf of a product

    I am trying to prove the following:
    \underbrace{(\lim \inf f_n)}_f\underbrace{(\lim\inf g_n)}_g\leq \underbrace{\lim\inf (f_ng_n)}_k.
    Here's what I have so far:
    FTSOC assume fg>k. We know that f_n<f-\epsilon and g_n<g-\epsilon for only finitely many n. Then f_ng_n<fg-\epsilon(f+g-\epsilon) for only finitely many n. This implies f_ng_n\geq fg-\epsilon(f+g-\epsilon) almost always. So, f_ng_n\geq k-\epsilon(f+g-\epsilon) almost always.
    But this doesn't get me anywhere. Where am I going wrong? If I let \epsilon=\min\{f,g\}, I just get f_ng_n\geq 0. So, this tells me nothing.
    Any help would be appreciated. (Even though it doesn't state it, I am assuming that all f_n, g_n are nonnegative.)
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  2. #2
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    I don't think that is true.
    Consider f_n the sequence -1,0,-1,0,...
    and g_n the sequence 2,0,2,0,...
    so f_ng_n is -2, 0,-2, 0,...

    Now (\lim \inf f_n)(\lim\inf g_n) = -1 \times 0 = 0
    \lim\inf (f_ng_n) = -2
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  3. #3
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    Quote Originally Posted by amheissan View Post
    ...I am assuming that all f_n, g_n are nonnegative.
    You are right that it wouldn't hold if were allowed to have a function as you have described, but we are limited to non-negative functions.

    Any more thoughts on proving this property? Thanks!
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by amheissan View Post
    You are right that it wouldn't hold if were allowed to have a function as you have described, but we are limited to non-negative functions.

    Any more thoughts on proving this property? Thanks!
    Note that by definition \displaystyle \liminf_{n\to\infty}f_n(x)=\inf \left\{\lim_{n\to\infty}f_{n_k}(x):f_{n_k}(x)\text  { is a subsequence of }f_n(x)\right\} and similarly for \displaystyle \liminf_{n\to\infty}g_n(x). Use the fact then that \inf\left(A\right)\inf\left(B\right)\leqslant \inf\left(AB\right).
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  5. #5
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    Okay, but how would you prove \inf(A)\inf(B)\leq \inf (AB)? I'm really just not seeing this property.
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  6. #6
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    For only nonnegative sequences, Drexel28 is giving you the hint to only look at the inf of the set of values at any subsequence (ignore the order of the sequence).

    Let AB = \{ab | a\in A, b\in B\}.
    Consider any element a\in A and b\in B, ab \geq \inf(A)\inf(B) (by definition since everything is nonnegative).

    Also, remember that the inf of a subset of AB must be larger than the inf of AB
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