(i) Show that (2, 5] considered as a metric subspace of $(\Bbb R,d)$ is not complete.
(ii) Show that [6,+∞) considered as a metric subspace of $(\Bbb R,d)$ is complete
So for (i) I think its not complete because its neither open nor closed and only closed sets can be complete
Then for (ii) I think its complete because I believe $[6, +\infty)$ is technically closed so therefore its complete.
Does that make sense, will that work? Just feel like its too little writing...
Yes, that works. I don't know why is it that you wrote that $(2,5]$ is not open. That is correct, of course, but irrelevant. What matters is that it is not closed. And $[6,+\infty)$ is simply closed, not “technically closed”, whatever that means.