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I'm stuck on how to show one of these is compact, and I want to verify my method for the other.

This one I am stuck on:

Proposition Let $(\mathbb{Q},d)$ be a metric space with $d(a,b)=|a-b|$. Let $E \subset \mathbb{R}$ and let $E=\{r \in \mathbb{Q}: r^2 < 2\}$. Show $E$ is not compact.

This one I believe I have solved and want to verify:

Proposition Let $\{x_n\}$ be a convergent sequence in $\mathbb{R}$ that converges to $x$. Let $E=\{x_n : n=1,2,3...\}\cup \{x\}$. Then $E$ is compact.

I believe I demonstrated this by the following. Since $\{x_n\}$ is convergent fix $\epsilon =1$ and then for $N \in \mathbb{N}$, there exists an $n \geq N$ such that $|x_n-x|<1$. Then take $\bar{\epsilon}=\max\{d(x_1,x)...d(x_N,x), 1\}$. Clearly $E \subset (\bigcup\limits_{k=1}^{N} N(x_k, \bar{\epsilon}))\bigcup N(x, 1))$ - this is a finite subcover of $N+1$ sets.

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For the first proposition, Heine-Borel Theorem asserts that every compact subset of $\mathbb{R}$ is closed and bounded. $E$ is clearly bounded, but is it closed? Can you find a limit point of $E$ that is not in $E$? Can you then find a Cauchy sequence in $E$ that does not have a limit in $E$?

The proof of the second proposition is incorrect. Why are you fixing $\epsilon$? The requirement for compactness is that whenever I give you a list of open sets whose union covers $E$, you can give me a finite subcover. I am not going to be so nice to only give you open sets expressible as open intervals of length $\bar{\epsilon}$....

Let me give you a hint: if $O$ is an open subset of $\mathbb{R}$ such that $x\in O$, then there are only finitely many of the $(x_n)$ that lies outside of $O$. (Firstly, why is that true? Secondly, how does this help you with the proof?)

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The first $N$ (finitely many) terms of a convergent sequence are more than $\epsilon$ apart. So my goal was to center a neighborhood around those and the limit point. So following your hint. Given $x \in O \subset \mathbb{R}$, this contains infinitely many points of the sequence. All that is left are the first $N$ terms. Could I just cover each one with an arbitrary open set (not defining a radius) centered at $x_n, n=1,2,...,N$? Also I noticed that my problem as stated is incorrect. I needed to show E is not compact. I have show that E is closed and bounded AND open. –  abet Oct 25 '12 at 7:51
    
@abet: again, you need to pick a finite subcover from the cover that I gave you. So you cannot "just cover each one with an arbitrary open set", because said "arbitrary open set" may not belong to the cover that I gave you to start with. Now, the question is: how do you go about choosing a collection of open sets from the cover that I gave you, so that every one of the first $n$ points lives inside their union? –  Willie Wong Oct 26 '12 at 7:52
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