# Math Help - Closed set problem

1. ## Closed set problem

Let n be in the set of Natural Numbers, let S = {u in R^n : ||u|| = 4}. Prove S is closed.

2. Originally Posted by tttcomrader
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