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Math Help - converge uniformly?

  1. #1
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    converge uniformly?

    On [a,b], the function sequence \{f_n\},\{g_n\} converge uniformly to f,g respectively. Suppose there exists positive sequence M_n such that f_n(x)\leq M_n, g_n(x)\leq M_n,\ \forall\ x\in [a,b]. Prove that f_ng_n converge unformly to fg on [a,b]

    PS: If M_n=M, I know how to prove. But this?....Would you help me?
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  2. #2
    Senior Member MacstersUndead's Avatar
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    Re: converge uniformly?

    To show that f_ng_n converges uniformly to fg on [a,b], we have to show there exists an N such that for all x in the interval [a,b] and n \geq N,

    |f_ng_n - fg|\leq \epsilon

    I'm sorry I'm a bit rusty, but perhaps it would help to note that you can consider an upper bound for your positive sequence M_n. If the least upper bound for M_n is a real number, say M, then you can say f_n \leq M and g_n \leq M for all n. If there is no least upper bound for your positive sequence, I'm not sure how to continue.
    Last edited by MacstersUndead; January 2nd 2013 at 11:37 PM.
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  3. #3
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    Re: converge uniformly?

    So what you're saying is that f_n and g_n are sequences of bounded functions that converge uniformly to f and g respectively on [a,b], and you want to prove that f_ng_n converges uniformly to fg.

    You said you know how to do it if f_n and g_n are uniformly bounded - that is, there are M_f and M_g such that |f_n(x)|<M_f on [a,b] and |g_n(x)|<M_g on [a,b]. So prove that first:

    Let N be such that |f_n(x)-f(x)|<\frac{1}{2} for all x\in[a,b]. Then for all n>N:

    |f_n(x)|\le |f_n(x)-f(x)|+|f(x)-f_N(x)|+|f_N(x)|\le \frac{1}{2}+\frac{1}{2}+M_N=M_N+1

    and I think you can probably fill in the rest of the proof.

    - Hollywood
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    Re: converge uniformly?

    Thank you very much indeed.
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