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Math Help - lim sup and lim inf proofs

  1. #1
    Senior Member Pinkk's Avatar
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    lim sup and lim inf proofs

    Prove that if k > 0 and (s_n) is a bounded sequence, lim sup ks_n = k * lim sup s_n and what happens if k<0?

    My attempt: If (s_n) is bounded, then lim sup s_n = S, where S is a real number, so lim sup (ks_n)= lim sup(k)(s_n)=S*lim sup(k) = S*k = k*lim sup s_n.

    Let lim inf s_n = L. If k<0, then lim sup (ks_n) = -lim inf(-k)(s_n)= -L*lim inf (-k) = -k *- L = k * lim inf (s_n).

    Is this correct?
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Pinkk View Post
    Prove that if k > 0 and (s_n) is a bounded sequence, lim sup ks_n = k * lim sup s_n and what happens if k<0?

    My attempt: If (s_n) is bounded, then lim sup s_n = S, where S is a real number, so lim sup (ks_n)= lim sup(k)(s_n)=S*lim sup(k) = S*k = k*lim sup s_n.

    Let lim inf s_n = L. If k<0, then lim sup (ks_n) = -lim inf(-k)(s_n)= -L*lim inf (-k) = -k *- L = k * lim inf (s_n).

    Is this correct?
    \sup_{m\leqslant n}\{k\cdot a_n\}=k\sup_{m\leqslant n}a_m
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  3. #3
    Senior Member Pinkk's Avatar
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    I'm not sure I follow. Was my proof incorrect?
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Pinkk View Post
    I'm not sure I follow. Was my proof incorrect?
    We have that \limsup\text{ }a_n=\lim_{n\to\infty}\{\sup_{n\leqslant m}a_m\} using the above see that \limsup\text{ }k\cdot a_n=\lim_{n\to\infty}\{\sup_{m\leqslant n}k\cdot a_m\}= \lim_{n\to\infty}\{k\cdot\{\sup_{m\leqslant m}a_m\}=k\lim_{n\to\infty}\{\sup_{m\leqslant n}a_m\}=k\limsup\text{ }a_n. I actually can't follow what you did :S
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  5. #5
    Senior Member Pinkk's Avatar
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    I used the property that, given a sequence (s_{n}) that converges to s and any other sequence (t_{n}), then lim\, sup\, (s_{n}t_{n}) = s \cdot lim\, sup\, (t_{n}). In my proof, I used the sequence (k), which obvious converges to k.
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  6. #6
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Pinkk View Post
    I used the property that, given a sequence (s_{n})that converges to s and any other sequence (t_{n}), then lim\, sup\, (s_{n}t_{n}) = s \dot lim\, sup\, (t_{n}). In my proof, I used the sequence (k), which obvious converges to k.
    Well then that is obviously correct, if you have already proved that theorem. But, that is killing a fly with an elephant gun in my opinion.
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  7. #7
    Senior Member Pinkk's Avatar
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    I guess it isn't a very good exercise then, since the general theorem was proven in the chapter before the exercises.
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