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Math Help - subsequences in R^n

  1. #1
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    subsequences in R^n

    i'm having difficulty proving the following:
    let p_i(i in naturals) be a sequence of pts in R^n and p_0 be a point in R^n. let p_i=(p_i,1 , ... , p_i,n) for i in naturals and p_0 = (p_0,1 , ... , p_0,n). show that lim(i goes to infinite) p_i = p_0 if and only if lim(i goes to infinite) p_i,k = p_0,k for every k=1,...,n.
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  2. #2
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    Quote Originally Posted by squarerootof2 View Post
    i'm having difficulty proving the following:
    let p_i(i in naturals) be a sequence of pts in R^n and p_0 be a point in R^n. let p_i=(p_i,1 , ... , p_i,n) for i in naturals and p_0 = (p_0,1 , ... , p_0,n). show that lim(i goes to infinite) p_i = p_0 if and only if lim(i goes to infinite) p_i,k = p_0,k for every k=1,...,n.
    I will do it for n=2 but it generalizes.

    Let \bold{x}_n = (a_n,b_n) and \bold{x}_0 = (a_0,b_0).

    If \bold{x}_n \to \bold{x}_0 then |(a_n,b_n) - (a_0,b_0)| can be made arbitrary small.
    Thus, |a_n-a_0|,|b_n-b_0|\leq \sqrt{(a_n-a_0)^2+(b_n-b_0)^2} < \epsilon.

    And if |a_n - a_0|,|b_n-b_0| < \epsilon then \sqrt{(a_n-a_0)^2+(b_n-b_0)^2} \leq \epsilon \sqrt{2}.
    Thus, \bold{x}_n \to \bold{x}_0.

    This is Mine 11th Post!!!
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