I took a real analysis course recently and now that the class is over, I decided to go back and read some of the sections we skipped over. Right now I'm reading "Some Topological concepts in Metric Spaces" in Ross's Elementary Analysis.
They explained that a metric space $(S,d)$, where $S$ is the space and $d$ is the the metric, is complete if every Cauchy sequence in $S$ converges to some element in $S$.
There is a problem at the end of the section that defines two metrics, $d_1$ and $d_2$ on $\mathbb{R}^k$ and asks to:
Show $d_1$ and $d_2$ are complete metrics on $\mathbb{R}^k$
I was wondering if there is a separate definition for a complete metric, or if the question is asking if $(\mathbb{R}^k, d_1)$ and $(\mathbb{R}^k, d_2)$ are complete.
If the latter is the case, is it proper language to ask if the metric itself is complete on a space rather than asking if the metric space is compete?
I realize this is probably a bit nit-picky, but I like to pay close attention to wording to avoid confusion.
A metric space is complete if every Cauchy sequence converges in the space. A metric is complete if the metric space it generates is complete.