I need to prove that if I := [0, 1/n], the element 0 belongs to all I(n) and the archimedian property to show that 0 is the only common point.(adsbygoogle = window.adsbygoogle || []).push({});

so basically, I need to prove the intersection from 1 to infinite of I(n) = {0}.

the book says to use the archimedian property that if t > 0, then tehre exsists n(t) such that 0 < 1/n(t) < t.

i've been thinking all day, but still can't go anywhere.

ok, so I guess to start off,

let t > 0, this implies that that there is a n in N, such that t < n(t).

so this means 1/n(t) < t.

and I guess by looking at the archimedian property that 0 < 1/n(t) < t, this means that 1/n(t) is always less then any given t. so I guess if it is less then t, then obviously, t is not included with any 1/(t + 1), and it is always greater then 0, so I guess that's why 0 is the only thing in common w/ all of them.

notice I said "I guess" a lot, because I really dont believe my proof......so can someone give me some tips? am I close?

and then, part 2:

same question, but with (0, 1/n) and need to prove that the intersection is the empty set. both are confusing

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# Basic question regarind archimedian property

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