# Find the $\bigcap_{n = 1}^{\infty} (-\frac{1}{n}, \frac{2}{n})$

Find the $\bigcap_{n = 1}^{\infty} (-\frac{1}{n}, \frac{2}{n})$

So the way I understand it is that I’m trying to find

$(\frac{-1}{1}, \frac{2}{1}) \bigcap (\frac{-1}{2}, \frac{2}{2}) \bigcap (\frac{-1}{3}, \frac{2}{3})\bigcap …$ and so forth. Then the intersection would be $\varnothing$ right? Can I prove this by just writing out the first few elements and seeing that they do not intersect?

Or is the question asking me to find:

$\frac{-1}{1} \bigcap \frac{2}{2} \bigcap \frac{-1}{3} \bigcap…$? in this case the intersection would still be $\varnothing$, right?

#### Solutions Collecting From Web of "Find the $\bigcap_{n = 1}^{\infty} (-\frac{1}{n}, \frac{2}{n})$"

There are three things to prove:

1. $0$ is in every interval, hence in the intersection of all of them.

2. If $x>0$, then there is at least one interval that does not contain $x$, so $x$ is not in the mutual intersection. (choose $n$ so that $2/n<x$).

3. If $x<0$, then there is at least one interval that does not contain $x$, so $x$ is not in the mutual intersection. (choose $n$ so that $-1/n>x$).

Combining, the intersection is $\{0\}$, as tetori points out.

Here is a longer and more formal version of vadim123’s answer, with some more detail about where the case split is coming from.


$$\calc x \in \langle \cap n :: (-\tfrac{1}{n}, \tfrac{2}{n}) \rangle \calcop\equiv{definition of \;\cap\;} \langle \forall n :: x \in (-\tfrac{1}{n}, \tfrac{2}{n}) \rangle \calcop\equiv{definition of interval; multiply by \;n\; — to try and isolate \;n\;} \langle \forall n :: -1 < n \times x < 2 \rangle \endcalc$$

Our strategy is to isolate $\;n\;$, and so we want to divide by $\;x\;$, and therefore we need to split into three different cases: for $\;x = 0\;$ we get

$$\calc \tag 1 \langle \forall n :: -1 < n \times x < 2 \rangle \calcop\equiv{substitute \;x = 0\;} \langle \forall n :: -1 < 0 < 2 \rangle \calcop\equiv{simplify} \true \endcalc$$

for $\;x > 0\;$ we get

$$\calc \tag 2 \langle \forall n :: -1 < n \times x < 2 \rangle \calcop\equiv{divide by \;x\;, using \;x > 0\; so no sign flip} \langle \forall n :: -\tfrac 1 x < n < \tfrac 2 x \rangle \calcop\Rightarrow{choose any \;n \geq \tfrac 2 x\;, possible since \;\mathbb N^+\; is unbounded upwards} \false \endcalc$$

and finally for $\;x < 0\;$ we get

$$\calc \tag 3 \langle \forall n :: -1 < n \times x < 2 \rangle \calcop\equiv{divide by \;x\;, using \;x < 0\; so the signs flip} \langle \forall n :: \tfrac 2 x < n < -\tfrac 1 x \rangle \calcop\Rightarrow{choose any \;n \geq -\tfrac 1 x\;, possible since \;\mathbb N^+\; is unbounded upwards} \false \endcalc$$

Summing up, we have proven that for all $\;x\;$, $\;x \in \langle \cap n :: (-\tfrac{1}{n}, \tfrac{2}{n}) \rangle \;\equiv\; x = 0\;$, in other words, the set in question is equal to $\;\{0\}\;$.

Note how the crucial property that we used was that $\;\mathbb N^+\;$ is unbounded.