# Thread: characteristic of closure and convergence

1. ## characteristic of closure and convergence

Again using this theorem: Let E be a set of real numbers. Then p is an accumulation point of E iff the following condition holds:
there is a sequence (xn) of members of E, each different from p, such that (xn) converges to p.

How would you prove or gain the following characterization about closure?

(1) Let E be a set of real numbers. A real number p belongs to the closure of E iff there is a sequence (xn) of members of E converging to p.

(2) Conclude, in particular, that if E is a non-empty bounded subset of R and s = supE, then there is an increasing sequence (xn) of members of E converging to s.

2. Originally Posted by dfanforlife
Again using this theorem: Let E be a set of real numbers. Then p is an accumulation point of E iff the following condition holds:
there is a sequence (xn) of members of E, each different from p, such that (xn) converges to p.

How would you prove or gain the following characterization about closure?

(1) Let E be a set of real numbers. A real number p belongs to the closure of E iff there is a sequence (xn) of members of E converging to p.

(2) Conclude, in particular, that if E is a non-empty bounded subset of R and s = supE, then there is an increasing sequence (xn) of members of E converging to s.
What have you tried? THe first part is just a consequence of your other question.

3. Note, however, that the words "different from p" do not appear in (i). That's important! dfanforlife, what is your definition of "closed"?

4. Originally Posted by HallsofIvy
Note, however, that the words "different from p" do not appear in (i). That's important! dfanforlife, what is your definition of "closed"?
My definition of closed..more so closure is: p is in the closure iff every nrighborhood of the point p contains an element of A. The part that confuses me is how to use the first condition or thm to get to the characteristic of closure stated in the original question. So I don't actually know how to use the thm listed to get the characterization in part 1. The for part 2 I'm confused as to whether or not I'm still using the thm and don't even know where to start.

5. Originally Posted by dfanforlife
My definition of closed..more so closure is: p is in the closure iff every nrighborhood of the point p contains an element of A.
That definition implies that $p\in\overline{A}$ means that $p\in A$ or else $p$ is an accumulation point of $A$.

6. Originally Posted by Plato
That definition implies that $p\in\overline{A}$ means that $p\in A$ or else $p$ is an accumulation point of $A$.
So for part (1) if the theroem holds, doesn't that imply that part (1) holds? If so how do I go the other way with the proof. Would I assume p is in the closure of E and then ...I think I would show that there is a sequence (xn) of members of E converging to p? <-if so, how?

7. Originally Posted by dfanforlife
If so how do I go the other way with the proof. Would I assume p is in the closure of E and then ...I think I would show that there is a sequence (xn) of members of E converging to p? <-if so, how?
That may not be true. If $p\in\overline{E}$ there may not be sequence of distinct points in $E$ that converges to $p$.

However, if $p$ is an accumulation point of $E$ there is such a sequence.