Originally Posted by

**zebra2147** To be honest we haven't talked about Cauchy sequences so I'm not sure how to connect all these.

However, I will do my best with what I have gathered form Wikipedia...

Since $\displaystyle x_0$ is a limit point in $\displaystyle D$ then for $\displaystyle x_0\in\overline{D}$ there must be a sequence of points that are contained in $\displaystyle D$ that converge to $\displaystyle x_0$.

Therefore, we have a sequence of points that are converging to a point $\displaystyle x_0$. Thus,(from what I gather about Cauchy Sequences), this sequence is a Cauchy sequence since the points are converging to $\displaystyle x_0$.

Then, (I got the following from Wikipedia) M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M or alternatively if every Cauchy sequence in M converges in M.

Here, our Cauchy sequences is converging to $\displaystyle x_0$ where $\displaystyle x_0\in \overline{D}$. So, $\displaystyle \overline{D}$ is complete?

Sorry this is might be kinda weak but I did what I could for never have learning about Cauchy Sequences or completeness....