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Thread: how can u prove this argument

  1. #1
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    how can u prove this argument

    $\displaystyle f_{n},g_{n}$ are sequences of functions :$\displaystyle X\rightarrow R$ which converge uniformly to f,g respectively.

    If $\displaystyle \lambda\in R $, prove $\displaystyle \lambda f_{n}$ converges uniformly to $\displaystyle \lambda f $

    and

    prove : $\displaystyle f_{n}g_{n} $ converges to $\displaystyle fg $ given f,g are bounded.
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  2. #2
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    Quote Originally Posted by szpengchao View Post
    $\displaystyle f_{n},g_{n}$ are sequences of functions :$\displaystyle X\rightarrow R$ which converge uniformly to f,g respectively.

    If $\displaystyle \lambda\in R $, prove $\displaystyle \lambda f_{n}$ converges uniformly to $\displaystyle \lambda f $

    and

    prove : $\displaystyle f_{n}g_{n} $ converges to $\displaystyle fg $ given f,g are bounded.
    You sent me a PM.
    I responded with a question: “What do you do for yourself?”
    Now I ask you the same question publicly.
    It seems to me that you simply expect to be given the answers.
    Am I wrong?
    If so, explain yourself!
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  3. #3
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    No. I have done part of this question.

    No. I done part of this question. I am not sure if my proof is right. and I am not sure my link between two arguments is right.

    i post my proof for the first part,


    when $\displaystyle \lambda=0$, question is trivial:
    $\displaystyle \epsilon_{1}>0, \ \forall x\in X, \ |\lambda f_{n}(x)-\lambda f(x)|=0<\epsilon_{1}$
    So assume $\displaystyle \lambda\neq0$
    $\displaystyle f_{n}$ converges uniformly to f:
    $\displaystyle \forall \epsilon_{1}>0, \ \exists N, \ \forall x\in X, \forall n\geq N, |f_{n}(x)-f(x)|<\epsilon_{1}$
    by writing $\displaystyle \epsilon_{2}=\frac{\epsilon_{1}}{|\lambda|}$:
    $\displaystyle \forall \epsilon_{2}>0, \ \exists N, \ \forall x\in X, \forall n\geq N, |\lambda f_{n}(x)-\lambda f(x)|<\epsilon_{2} \ \ \square.$
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  4. #4
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    and then

    and then i just said, by setting:
    $\displaystyle \lambda=g_{n}(x) \forall x\in X$
    and use lemma just proved,
    I got:
    $\displaystyle \lambda f_{n}(x)$ converges uniformly to $\displaystyle
    \lambda f$
    and then set $\displaystyle \lambda=f(x) $, use lemma again,
    $\displaystyle g_{n}(x) f(x) $ converges pointwise to $\displaystyle fg $
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  5. #5
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    ok

    ok. i posted all the proof i can get. so, could you please give me any correction of my proof if there is any problem.
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