Find $\bigcap_{n=1}^\infty(0,1/n)=\emptyset$

Question

I`ve tried this and is it true or completely not? Then how can I fix it?

Proof:

too wrong so I get it off

Answer 1

$$ \bigcap_{n>0}\left(0,\frac 1n\right) $$is a subset of $(0,1)$. For each $x\in (0,1)$ there is some $n>0$ ($n>1/x$), such as $$ \frac 1n < x $$ and for such $n$, $$ x\notin \left(0,\frac 1n\right) $$so $$ \bigcap_{n>0}\left(0,\frac 1n\right) = \emptyset $$

Answer 2

If you want to prove (but as other have said in comments to your question, your notation is quite poor, so I'm just guessing): $$ \bigcap_{n=1}^{\infty}]0,\frac{1}{n}[ = \emptyset $$ it's just a matter of finding an interval in the collection on the left that doesn't contain $x$ for any $x$.

Answer 3

Let's back up...

inf ⋂(0,1/n=∞)

Makes no sense. That notation means nothing.

• $\cap$ means "the intersection of the following sets."
• $(0,m)$ means $\{x : 0 < x < m\}$.
• When $\infty$ appears in the intersection, it has a specific meaning. It would actually be better to say $\bigcap_{n \in \mathbb{N}} (0, 1/n)$. Intersection means "and", so it would be better to write this as $\{x:x \in (0,1/n) \forall n \in \mathbb{N}\}$.

I did the hard part for you. I rigorously wrote down the actual set this question is talking about. You, on the other hand, just manipulated $\infty$ willy-nilly until you happened to write down what you were trying to prove. You don't know whether you took some random number of random steps and got the right answer, or whether your steps were correct. Please work rigorously from this definition, and understand how this definition makes sense. It's better to say "I don't know what this means" than to try proving it before you even think about what it means.