I want to prove that given a limit point, it is possible to find a sequence that converges to it

Question

Given this definition, Prove carefully that if $E \subset X$ and if $x$ is a limit point of $E$, then $\exists$ a sequence $(a_n)$ in $E$ that converges to $x$.

Proof: (attempt)

Let x be limit point of $E$. Let $r > 0$ be given and put $V = N_{r} (x)$

Notice that we can find $a_1 \in N_1(x)$ (putting $r = 1$) such that $a_1 \neq x$

We can find $a_2 \in N_{1/2} (x) $ such that $a_2 \neq x$

and so on. In general, we can find $a_n \in N_{1/n} (x) $ such that $a_n \neq x$ and $a_n \in E$.

Now, Let $\varepsilon > 0$ be given. We can find $N > \dfrac{1}{\varepsilon }$ so that for all $n > N$, we have

$$ d(a_n,x) < \dfrac{1}{n} < \dfrac{1}{N} < \varepsilon $$

so $a_n \to x$.

Is this a correct proof?

Answer 1

Yes, it is correct, but:

• The sentence “Let $r>0$ be given and put $V=N_r(x)$.” is useless. Neither $r$ nor $V$ are used again.
• There is no need to write that each $a_n$ is distinct from $x$. Yes, you can choose $a_n$ such that that occurs, but that's irrelevant for the proof.