Question

\[\bigcap^∞_{n=1}A_n=\{x|(∀n)(x\in A_n)\}\]

Answer 1

What is it, intersection of an infinite set? (integers?)

Empty, I guess...

Answer 2

\(A_n=n\)
\[\bigcap\limits^\infty_{n=1}n=\{x|(\forall n)(x\in n )\} =\{ x|1\cap2\cap3\cap\dots\}=\emptyset\]

Answer 3

\(A_n=11\)
\[\bigcap\limits^\infty_{n=1}A_n=\{x|11\cap11\cap11\cap\dots)\} =\{11\}\]

Answer 4

\[\cap_{n=1}^{\infty}A_n=\{x:\forall n,x\in A_n\}\]

Answer 5

\[\bigcap\limits^\infty_{n=1}n=\{x|(\forall n)(x\in n )\} =\{ x|1\cap2\cap3\cap\dots\}=\emptyset\]doesn't make sense. if \(A_n=\{n\}\) the set with one element, then
\[\bigcap\limits^\infty_{n=1}n=\{x|(\forall n)(x=n )\} =\emptyset\] you cannot take the intersection of numbers, only sets

Answer 6

does this make sense then?\[\bigcap\limits^\infty_{n=1}n=\{x|(\forall n)(x\in n )\} =\{ 1\}\cap\{2\}\cap\{3\}\cap\dots=\emptyset\]