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Math Help - Pointwise convergence help

  1. #1
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    Pointwise convergence help

    ok guys i'm trying to get my head around pointwise convergence.

    I'm working through an example, I = [0,1]

    fn(x) = 0 ,for x=0
    = ((-1)^n)n ,for x E (0,1/n]
    = 0 ,for x E (1/n,1]


    The answer:
    pick some N large enough so that it satisfies 1/N < x. Then fn(x) = 0 for all n less than or equal to N and so fn(x) ---> 0


    I can't seem to see why how picking an N proves the convergence?

    any help would be much appreciated
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  2. #2
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    Quote Originally Posted by maths52 View Post
    ok guys i'm trying to get my head around pointwise convergence.

    I'm working through an example, I = [0,1]

    fn(x) = 0 ,for x=0
    = ((-1)^n)n ,for x E (0,1/n]
    = 0 ,for x E (1/n,1]


    The answer:
    pick some N large enough so that it satisfies 1/N < x. Then fn(x) = 0 for all n less than or equal to N and so fn(x) ---> 0


    I can't seem to see why how picking an N proves the convergence?

    any help would be much appreciated

    The puppy is indexed by n. thus that is what you're looking to make a statement about. everything in this problem centers around what n you're looking at. Now by choosing an N, you're taking a step toward putting a condition on n. Without putting a condition on n, how can you prove convergence when the sequence of functions f only have meaning in relation to n. So you restrict the statement to those functions that have n \geq N. By doing so, you guarantee that x>\frac1{N}>\frac1{n}. In those cases, f_n converges to 0.
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