# Can't understand a proof in Rudin

• kostas230
In summary, Rudin's proof shows that if x is not in \bar{E} then there exists a neighborhood N_r (x) that does not contain x.

#### kostas230

I can't understand a proof in the Theorem 2.27, part (a).

If $X$ is a metric space, $E$ a subspace of X, and $E'$ the set of all limit points of $E$, we denote by $\bar{E}$ the set: $\bar{E}=E \cup E'$

We need to prove that $\bar{E}$ is a closed set. Rudin's proof is this:

If $x\in X$ and $x \notin \bar{E}$ then $p$ is neither a point of $E$ nor a limit point. Hence, $p$ has a neighborhood which does not intersect $E$. Therefore, the complement of $\bar{E}$ is closed.

My question is this: How do we prove that there exists a neighborhood that does not contain any limit points of $E$?

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If every open ball at x contains a limit point of E, x would be a limit point of those limit points. Can you prove that it is contradictory for an open ball at x to be disjoint from E?

Edit: I changed "neighborhood" to "open ball", I think neighborhood is wrong although Rudin may define it differently.

I found a more straightforward proof. Suppose $p$ is a limit point of $E'$. Then, for every neighborhood $N_r (p)$ there exists a limit point $q$ of $E$. There exists a real number $h>0$ with $d(p,q)=r-h$. Consider the neighborhood $N_h (q)$

$d(s,p) \leqslant d(p,q)+d(q,s) < r$

Hence, $s\in N_r (p)$. Therefore, $p$ is a limit point of $E$.
Did I get it right? xD
(BTW, I should mention that I'm a physics undergrad; no training in rigorous math except a linear algebra self study with Axler. So, I'm a little slow in this. xD)

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That works. It is more technical than what I was alluding to above but it's fine.

You should probably mention that s is a point in E.

1 person

I would approach this question by first understanding the definitions and concepts involved. In this case, we need to understand what a limit point is and how it relates to neighborhoods. A limit point of a set E is a point p for which every neighborhood of p contains a point of E other than p itself. In other words, p is a point where E "accumulates" or "approaches" in some sense.

Now, in order to prove that there exists a neighborhood of x that does not contain any limit points of E, we can use the fact that x is not a limit point of E. This means that there exists a neighborhood of x that does not intersect E. Since this neighborhood does not intersect E, it also does not contain any limit points of E. This is because if it did contain a limit point of E, then it would also intersect E.

Furthermore, since x is not a limit point of E, this means that there exists a positive distance between x and any point in E. This distance can be used to define a neighborhood of x that does not contain any limit points of E. For example, we can take a neighborhood with radius smaller than this distance. This neighborhood will only contain points that are a positive distance away from x, and therefore cannot be limit points of E.

In summary, we can prove that there exists a neighborhood that does not contain any limit points of E by using the fact that x is not a limit point of E and defining a neighborhood with a radius smaller than the distance between x and any point in E. This neighborhood will not intersect E and therefore will not contain any limit points of E.

## 1. Why is understanding proofs in Rudin difficult?

Understanding proofs in Rudin can be difficult because Rudin's mathematical writing is very concise and formal, and often assumes a strong background in mathematical concepts and notation. This can be challenging for those who are not yet comfortable with these concepts.

## 2. How can I improve my understanding of proofs in Rudin?

To improve your understanding of proofs in Rudin, it is important to have a strong foundation in mathematical concepts and notation. It may also be helpful to read and study the proofs carefully, and to consult other resources or seek help from a math tutor or professor.

## 3. What should I do if I can't understand a specific proof in Rudin?

If you are struggling to understand a specific proof in Rudin, try breaking it down into smaller parts and understanding each step individually. You can also consult other resources, such as online forums or textbooks, or seek help from a math tutor or professor.

## 4. Is it necessary to understand every proof in Rudin?

It is not necessarily necessary to understand every proof in Rudin, as long as you have a general understanding of the main concepts and ideas presented. However, if you are studying for a math exam or pursuing a career in mathematics, it is important to have a thorough understanding of the proofs in Rudin.

## 5. How can I overcome the fear of not understanding proofs in Rudin?

To overcome the fear of not understanding proofs in Rudin, it may be helpful to remind yourself that it is a natural and expected part of the learning process. It can also be helpful to break down the proofs into smaller parts and seek help from a math tutor or professor if needed.