The set of rationals Q forms a metric spce by
Then a subset E of Q is defined by
So I am trying to show that E is closed and bounded, but not compact.
To me, it is clear than E is bounded (by 2 and 3?!).
I am having trouble showing E is closed. I believe 2 and 3 are limit points of E but E does not contain them, so E is not closed? also, E does not contain e = 2.718281828.... but that is also a limit point of E, is it not? Maybe I am confused with my definition of limit point.... are any of these (2, 3, e) actually limit points of E?
I think I can show E is not compact, since the open cover of E,
for n = 2, 3, 4, ....
has no finite subcover?
I am worried I am perhaps misunderstanding some definitions... Any help or direction would be greatly appriciated.
Thank you in advanced for you time!