Convergent and Cauchy sequences in metric spaces

Question

I understand that:

In any metric space X, every convergent sequence $\{p_n\}$ is a Cauchy sequence, which can be shown by noting that, for all $\epsilon >0$, there exists an integer $N$ such that $n,m \geq N$ implies

$d(p_n,p_m) \leq d(p_n,p) + d(p, p_m) < 2\epsilon$.

Since $\epsilon$ was arbitrary, we conclude that $\{p_n\}$ is Cauchy. But also, it is true that in $R^k$, which is also a metric space, every Cauchy sequence converges (which is a little bit harder to prove).

I have a couple questions:

1. Following from the statements above, can we say WLOG, a sequence $\{p_n\}$ in $R^k$ is convergent if and only if it is a Cauchy sequence?
2. Also, I don't think I could infer that the statement is true in every metric space. Is there a counterexample to see this?

Thanks!

Answer 1

In $\mathbb{R}^n$, every Cauchy sequence converges. This is a property called completeness; a metric space $X$ is complete if every Cauchy sequence converges. Thus, in a complete metric space, which $\mathbb{R}^n$ is, a sequence is Cauchy if and only if it converges.

For your second question, just take a non-complete metric space, say, $\mathbb{Q} \subset \mathbb{R}$, and consider a sequence of rational numbers that are converging to $\sqrt{2}$ in $\mathbb{R}$. Since $\sqrt{2}$ is not a rational number, this sequence is Cauchy, but it does not converge in $\mathbb{Q}$.