Very elementary question concerning open intervals and their intersection Let $I_n = (a_n, b_n ) $ be  open intervals in $\mathbb{R}$. I am triyng to show that $ \bigcap_{1 \leq n \leq k} I_n $ is open set.
sol attempt
let $x \in \bigcap_{n=1}^k I_n $. Then, $ x \in I_n$ for all $n$. In particular, we can find $r_1,...,r_k > 0$ such that
$$ (x - r_i, x + r_i ) \subseteq  I_i$$
for $i = 1,2,...,k$. Let $ r = \min \{ r_1,...,r_k \} $. Then, notice
$$ x -r_i \leq x - r < z < x + r \leq x + r_i $$
for $i = 1,2,...,k$. In other words, we have that
$$ z \in (x-r,x+r) \implies z \in I_i \; \; \; \text{for all} \; \; i $$
Hence, $$ (x -r,x+r) \subseteq \bigcap I_i $$. In other words, the intersection of open intervals is open.
I would love to enjoy some feedback. Thanks.
 A: Your proof is correct.
In fact, it is possible to show that $$\bigcap_{1\leq n\leq k}I_n$$
is not only an open set but an interval. You can show this by showing $J_1\cap J_2$ is an interval for any two intervals $J_1,J_2$, then using induction:


*

*For $k=1$, $\bigcap_{1\leq n\leq 1}I_n = I_1$ is clearly an interval.

*Using the fact that $\bigcap_{1\leq n\leq k}I_k$, it is easy to see that $$\bigcap_{1\leq n\leq k+1}I_n = \bigcap_{1\leq n\leq k}I_n\cap I_{k+1}$$ is an intersection of two intervals and thus an interval.

A: This is actually true for the intersection of finitely many open sets in any metric space - not just in $\mathbb{R}$.  Point being, it's almost easier to prove the general case.  The proof is straightforward:
Let $\displaystyle S = \bigcap_{k = 1}^n A_k$, where the $A_k$'s are open sets. Choose any $x \in S$.  Then, for every $A_k$, there is an $r_k$ such that $N_{r_k}(x) \subset A_k$.  So simply choose $r = \min(r_k)$, and we have $\displaystyle N_r(x) \subset S$.  Thus, $S$ is an open set.
This is almost exactly what you did above.  However, working in the general space helped us get rid of a bit of extra notation that comes along with working in $\mathbb{R}$.

A word of caution: This result is true only when we intersect finitely many open sets.  It is possible for the intersection of infinitely many open sets to be closed.  For example, consider $\displaystyle \bigcap_{n = 1}^\infty \left(- \frac{1}{n}, \frac{1}{n}\right)$.
