A Cauchy sequence is convergent over the complex plane.

I'm trying to prove that a Cauchy sequence is convergent over the complex plane. I think I have proved it, yet I would like to be sure. Any aid is greatly appreciated. Here's what I've done:

Let $\{z_n\}$ be a Cauchy sequence in $\mathbb{C} \Rightarrow \forall \varepsilon > 0 \quad \exists N \in \mathbb{N}$ such that $\forall n,m \geq N \Rightarrow ||z_n - z_m|| < \varepsilon$. Then $z_m \in B_{\varepsilon} (z_n)$. We have that $B_{\varepsilon} (z_n) \subset \overline{B_{\varepsilon} (z_n)}$. Since $\overline{B_{\varepsilon} (z_n)}$ is compact and $B_{\varepsilon} (z_n)$ has infinitely many points, then it must have a limit point, making $\{z_n\}$ convergent.

• I don't see where you've fixed an $n$ – anonymous Aug 24 '17 at 0:36
• You mean, the point I chose to be the center of the ball I proposed? I haven't thought of that, yet let's say I do fix said center, because $n,m \geq N$ then all succesive points in the sequence must fall inside that ball, right? Yet I'm starting to think that my proof fails when I consider all the ball and not just the points of the sequence, because maybe I can have more than just one limit point. (I deleted my comment and reposted because the site wouldn't let me edit my old comment and it had a mistake). – user367420 Aug 24 '17 at 2:59

The correct way to prove it is:

You can prove that every Cauchy sequence is bounded so $z_n$ is bounded.

Thus exists $M>0$ such that $$||z_n-0||=||z_n|| \leq M,\forall n \in \mathbb{N} \Rightarrow z_n \in cl(B(0,M))$$

Then you use your following arguments with the limit point compactness.

Here it is another proof if you want to take a look at:

Let $z_n \in \mathbb{C}$ a Cauchy sequence.

$z_n=x_n+iy_n$

We have that $$||z_m-z_n|| \rightarrow 0 \Rightarrow \sqrt{(x_n-x_m)^2+(y_n-y_m)^2} \rightarrow 0$$ as $m,n \rightarrow + \infty$

We have that $$|x_n-x_m|=\sqrt{(x_n-x_m)^2} \leq \sqrt{(x_n-x_m)^2+(y_n-y_m)^2} \rightarrow 0$$ $$|y_n-y_m|=\sqrt{(y_n-y_m)^2} \leq \sqrt{(x_n-x_m)^2+(y_n-y_m)^2} \rightarrow 0$$

Thus $x_n,y_n$ are Cauchy sequences in $\mathbb{R}$ which is complete(i.e every Cauchy sequence in $\mathbb{R}$ converges).

So exists $x_0,y_0 \in \mathbb{R}$ such that $$x_n \rightarrow x_0$$ $$y_n \rightarrow y_0$$

Thus $z_n \rightarrow x_0+i y_0$ proving that $z_n$ converges.

• Just one question, why does $z_n$ must lie on the ball's border? I know it can because it is bounded, but why is it necessary? Thank you! – user367420 Aug 24 '17 at 2:51
• @R.Mor....According to the definition of a bounded sequence it is possible for $z_n$ to lie on the border, why not? Also you have $\leq$ in the definition...so maybe some terms are strictly smaller and even if all of the terms are striclty smaller than the radious $M$,in this case still the sequence is bounded..The nessecery part here is to bound a sequence in a compact set..It is not nessecery that the whole sequence lies in the border.Take for instance $\frac{(-1)^n}{n}$ in $[-1,1]$ – Marios Gretsas Aug 24 '17 at 12:10
• Right, I see it now. Thank you for the additional proof, too. Both were very helpful. – user367420 Aug 25 '17 at 1:25