Let $A = ]0,1]$ Find and open cover with no finite sub-cover

Question

Let $A = ]0,1]$ Find and open cover with no finite sub-cover.

I had constructed open cover such as $\{]1/n, 2[ :n = 1,2,3,...\}$ but can't be sure of whether this cover actually contain all of $A$. How could I prove that A actually contain all of $A$?

@JorgeFernándezHidalgo I think $]0,1]$ is French (or something) for $(0,1].$ — bof, Jun 02 '17 at 04:52
@Daschin it helps in proving that if $x>0$ then there is an integer $n>0$ with $1/n>x$. — Angina Seng, Jun 02 '17 at 04:55
@bof wow, this is the most horrible thing I have seen today. — Asinomás, Jun 02 '17 at 04:55
You need to find $n$ such that $\frac1n\lt\varepsilon,$ right? $\frac1n\lt\varepsilon$ is the same as $n\gt\frac1\varepsilon.$ Use Archimedes. — bof, Jun 02 '17 at 04:55
Actually, ]0,1[ vs. (0,1) has an advantage that it can not be confused for an ordered pair. I've seen it a lot on this sight. It looks odd to me but it's a bit much to call it "horrible". — fleablood, Jun 02 '17 at 05:24

score 1 · Answer 1 · answered Jun 02 '17 at 04:54

1

Consider the cover $A=]\frac{1}{n},1]$.

Suppose $B$ is a finite subset of $A$. Let $N$ be the largest value such that $]\frac{1}{N},1]$ is in $B$, then $\frac{1}{2N}$ is not in any element of $B$, thus $B$ does not cover $]0,1]$.

answered Jun 02 '17 at 04:54

Asinomás

105,651

score 0 · Answer 2 · answered Jun 02 '17 at 05:33

Your open cover does cover all $]0,1[$.

Let $x \in ]0,1[$ then $0 < x < 1$. Let $n$ be a positive integer so that $n > \frac 1x$. (The integers are not bounded so there will be such an integer-- this is actually one of the consequences of the Archimedean Property; it's something that actually has to be proven but... that's for another post.)

So $n > \frac 1x$ so $0 < \frac 1n < x$. So $x \in ]\frac 1n, 2[$. So all points $x$ in $A$ are covered by your subcover.

And I assume you figured out that now finite subset of this cover successfully covers? (as per Jorge Fernandez Hidalgo's answer shows.)

Let $A = ]0,1]$ Find and open cover with no finite sub-cover

2 Answers2