show that the metric space = { + : , ∈ ℚ} with usual metric is incomplete

Question

Let (,d) be a metric space with = { + : , ∈ ℚ} and d( , ) = |x − y|. Show that (,d) is incomplete I haven't been able to find any Cauchy sequence that is divergent in X other than this one: x1= 1 and xn+1= 2*(1 +xn)/(2 +xn), for n≥2 ,which is a sequence in ℚ, but it's also in since q=0 IS there any Cauchy sequence for which it diverges but with q≠0

Answer 1

Any Cauchy sequence in $X$ that converges in $\Bbb C$ to some $a+bi$ with at least one of $a$ and $b$ irrational must fail to converge in $X$, and there are lots of those. Let $\alpha$ be any irrational number, and let $\langle r_n:n\in\Bbb N\rangle$ be a sequence of rational numbers converging to $\alpha$. It’s not hard to show that $\langle r_n+0i:n\in\Bbb N\rangle$, $\langle 0+r_ni:n\in\Bbb N\rangle$, and $\langle r_n+r_ni:n\in\Bbb N\rangle$ are all Cauchy sequences in $X$ that do not converge in $X$.

And if $\beta$ is also irrational, and $\langle s_n:n\in\Bbb N\rangle$ is a sequence of rationals converging to $\beta$, then $\langle r_n+s_ni:n\in\Bbb N\rangle$ is yet another such sequence.