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I am trying to figure out how to write out the answer to this. If I am given: $$A_i = \{i,i+1,i+2,...\}$$ And I am trying to find the intersection of a collection of those sets given by: $$\bigcap_{i=1}^\infty A_i $$

First off, am I right in saying its the nothing since: $$A_1 = \{ 1,2,3,...\}$$ $$A_2 = \{ 2,3,4,...\}$$

So, would the answer be the empty set?

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Yes, that's correct, the answer is the empty set. To explain this properly you want to take a number $n \in \mathbb N$ and explain why $n \notin \bigcap A_i$. If you do this with no conditions on $n$ you will have shown that the intersection doesn't contain any positive integers, hence it's empty.

To show $n \notin \bigcap A_i$ you must pick an $i$ such that $n \notin A_i$. I leave it to you to determine which $i$ to pick.

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Choose $n$, and note that $\cap_{k=1}^{n+1} A_k = A_{n+1}$, and $n \notin A_{n+1}$. Since $\cap_k A_k \subset A_{n+1}$, we see that $n \notin \cap_k A_k$. Hence $\cap_k A_k = \emptyset$.

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