## Background

To prove a limit, we need to find a number before. The definition
gives us *no* hints on how to find it. Cauchy’s Criterion for
convergence is a good method for finding it numerically in “certain
spaces” (e.g. $\R,L^p,L^\infty$) because the definition is
*intrinsic*.

Unluckily, there’re some “strange spaces” where Cauchy’s Criterion
*isn’t* enough to give a convergence. The simplest one is $\Q$
equipped with the usual metric in $\R$.

One knows that an incomplete metric space $X$ can extended to form a complete one $X^*$ (which is unique up to isometric isomorphism) so that $\iota(X)$ is dense in $X^*$. That is, $\overline{\iota(X)} = X^*$.

## Problem

I confused $\overline{X}$ with $X^*$. In other words, what’s the difference between the completion of $X$ and the closure of $X$?

## Explanation

### General answer

To see this, we note that the closedness of a set is discussed
*relative to another set*, so if one stays inside the incomplete space
$X$, then one notices that **$X$ is closed in $X$**. However, even
though one *doesn’t* know about $X^*$, one still realises that $X$ is
incomplete.

If one becomes wiser and seeks to “complete a given Cauchy sequence”
(i.e. find a limit for a Cauchy sequence in $X$), it’s *impossible to
live inside $X$*—one needs to escape from $X$ to $X^*$ so as to
enjoy more freedom.

Claim: $X$

isn’tclosed in $X^*$.Approach: We

won’tstart from the topological definition (open complement). Instead, we’ll argue with sequences.Proof: Since $X$ is

incomplete, there exists a Cauchy sequence $(x_n)$ in $X$ so that it hasnolimit in $X$. The completeness of $X^*$ is applied to construct a limit $x$ in $X^*$. If one allows $X$ to be closed, then $x$ will stay inside $X$, which contradicts the fact that $(x_n)$ hasnolimit in $X$.

Even though living in $X$ is sad, and it seems to be hard to escape
(because $X$ is closed in $X$), if one has a chance to travel outside,
one will see that $X$ is *not* really closed—it is possible to find
a sequence $(x_n)$ to escape from $X$.

### Specific example

$\Q$ is closed in $\Q$, but it *isn’t* closed in $\R$. Therefore, the
concept of closedness depends on the bigger space. However, the
concept of completeness only depends on $\Q$ itself.