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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- Mar 18th 2007, 07:22 PMtttcomraderClosed set problem
Let n be in the set of Natural Numbers, let S = {u in R^n : ||u|| = 4}. Prove S is closed.

- Mar 18th 2007, 10:48 PMCaptainBlack
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