# Questioning dense subset completeness (counterexample)

by LoneBone   Last Updated June 13, 2019 12:20 PM

Let $$X$$ be a separable metric space and $$A \subset$$ X be countable and dense. Characterize the statements below as true or false (and why).

• If every Cauchy sequence in $$A$$ converges in $$X$$, $$A$$ is complete.

• If every Cauchy sequence in $$A$$ converges in $$X$$, $$X$$ is complete.

• Is $$X = \mathbb{R} \setminus \{\sqrt{2}\}$$ and $$A = \mathbb{Q}$$ a direct counterexample of the two above statements?

For the first statement, I would say that it's false, since every Cauchy sequence of $$A$$ does not converge in $$A$$, so $$A$$ is not necessarily complete.

As for the second, there may be divergent Cauchy sequences of $$X$$, so false again.

Is my approach correct and if so, does the third statement play the role of a counterexample of the aforementioned?

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The first statement is “If every Cauchy sequence in $$A$$ converges in $$X$$, $$A$$ is complete.” You cannot deduce from this that not every Cauchy sequence of $$A$$ converges in $$A$$. What If it turns out that $$X=A$$?

The second stament is “If every Cauchy sequence in $$A$$ converges in $$X$$, $$X$$ is complete.” Concerning this, what you do is to claim that there may be divergent Cauchy sequences of $$X$$. That's wrong. There is no such sequence.

And $$X=\mathbb R\setminus\left\{\sqrt2\right\}$$ and $$A=\mathbb Q$$ is a counterexample to the first statment, but not to the second one: $$\left(\frac{\left\lfloor\sqrt2n\right\rfloor}n\right)_{n\in\mathbb N}$$ is a Cauchy sequence of elements of $$\mathbb Q$$, but it does not converge in $$\mathbb R\setminus\left\{\sqrt2\right\}$$.

José Carlos Santos
June 13, 2019 11:33 AM

For the first statement $$A=\mathbb Q$$, $$X=\mathbb R$$ gives a counterexample.

The second statement is true. Proof: let $$\{x_n\}$$ be a Cauchy sequence in $$X$$. For each $$n$$ there exists $$y_n \in A$$ such that $$d(x_n,y_n) <\frac 1 n$$. Now it is easy to verify (by triangle inequality) that $$\{y_n\}$$ is a Cauchy sequence in $$A$$. By assumption $$y_n$$ converges to some $$y$$ and triangle in equality helps you again to show that $$x_n \to y$$. Hence $$X$$ is complete. Obviously the third statement is false.

Kavi Rama Murthy
June 13, 2019 12:15 PM

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