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