What is $\bigcap_{i \in \Bbb N}\{i, i + 1, i + 2\}$?

Question

$$\bigcap_{i \in \Bbb N}\{i, i + 1, i + 2\}$$

So in this question, my teacher ask us to determine the set. But shouldn't the answer depend on the number of sets? If the sets if infinity, then there is no intersection. But if there are only three sets, where $i=1,2,3$. then there can be intersection?

I am really confused.

Answer 1

The set is empty. You have an infinite intersection of the sets $\{ i,i+1,i+2\}$ where $i \in \mathbb{N}$.

If $i=1$, then we have the set $\{ 0,1,2\}$. If $i=3$, then we have $\{ 3,4,5\}$. Since $\{ 0,1,2\}\cap \{ 3,4,5\}=\emptyset$, the whole intersection is empty.