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Math Help - Closed set problem

  1. #1
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    Closed set problem

    Let n be in the set of Natural Numbers, let S = {u in R^n : ||u|| = 4}. Prove S is closed.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by tttcomrader View Post
    Let n be in the set of Natural Numbers, let S = {u in R^n : ||u|| = 4}. Prove S is closed.
    Here a set is closed if all limit points of the set are in the set.

    Let x be a limit point of S, then there exists a convergent sequence x(n) n=1, 2, .., x(n) in S,
    which converges to x.

    So for all e>0, there exists a natural number N such that for all n>N

    ll x(n) - x ll < e.

    ll x(n) - x ll >= | llx(n)ll - llxll | = | 4 - llxll .

    so:

    | 4 - llxll | <= ll x(n) - x ll < e

    so:

    4-e < llxll < 4+e,

    hence as e>0 is arbitary llxll differes from 4 by less than every positive
    number, hence it is 4, and x is in S.

    RonL
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