Question

. show that a subset of rn is complete if and only if it is closed.

Answer 1

A subset of R^n is complete if and only if it is closed. This can be shown by proving that a complete subset contains all its limit points and is therefore closed.

Step-by-step explanation:

A subset of Rn is complete if and only if it is closed.

Proof:

1. Suppose the subset A of Rn is complete and let x be a limit point of A.
2. To show that A is closed, we need to prove that x is also an element of A.
3. Since x is a limit point of A, there exists a sequence {an} in A that converges to x.
4. As A is complete, the sequence {an} converges to a point a in A.
5. Since the limit of a convergent sequence is unique, we have x = a, which implies that x is an element of A.
6. Therefore, A contains all its limit points, and hence is closed.

Similarly, if a subset A of Rn is closed, we can show that it is complete using the same proof technique in the reverse direction.