Proving a space is complete by assuming Cauchy sequences converge

metric-spacesreal-analysis

I had an interesting debate with some classmates on how to prove that a space is complete.

I went through the process of taking a Cauchy sequence, finding a candidate limit, and showing that the limit is both in the space and is the actual limit of the Cauchy sequence.

However, some of my classmates instead supposed that the Cauchy sequence converged to something, and proceeded to show that the limit is in the space.

Is their method correct? It seems odd to me to assume that the Cauchy sequence converges. I know we could use this to show that the space is closed, but we are not assuming that the ambient space is complete.

They argue that every Cauchy sequence converges in the completion of the metric space so it is no problem assuming that the Cauchy sequence converges: all that is left is to show that the limit belongs to the space. Is this a legitimate strategy for showing that a space is complete?

EDIT: More explicitly, to show a space $X$ is complete, can we say “let $(x_n)$ be a Cauchy sequence that converges to $x$. It suffices to show $x\in X$?''

Best Answer

Let's try to figure out what exactly are they trying to do.

Every metric space $X$ has a completion $\overline X$, which is defined using equivalence class of Cauchy sequences.

In particular, if $\{x_n\}$ is the given Cauchy sequence, then the limit in the completion is .... $[\{x_n\}]$, the equivalence class in the completion.

One has the embedding $$\phi: X \to \overline X$$ given by $\phi(x) = [\{x, x, \cdots, x, \cdots\}]$ (the equivalence class of the constant Cauchy sequence), then to say that the limit is in $X$, it really means $$ [\{x_n\}] = [\{x, x, \cdots, x, \cdots\}]$$ in $\overline X$. Writing down the definition of the equivalence, it becomes:

for all $\epsilon >0$, there is $N\in \mathbb N$ so that $$ |x_n - x|<\epsilon$$ for all $n\ge N$.

(it would be $|x_n - y_n|<\epsilon$ for two sequence in general, but now one of them is constant).

This is exactly the definition of the convergence of $\{x_n\}$ to $x$.

Thus your classmate is not wrong strictly speaking. However, it seems to be a pointless complications. What's more, I don't see how thinking this way would help guessing the limit.

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