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proof9 .pdf


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9. Given an infinite collection An,n=1,2,… of intervals of the real line,
their intersection is defined to be ⋂ n=1∞ An={x|(∀n)(x∈An)} Give an
example of a family of intervals An,n=1,2,…, such that An+1⊂An for all n
and ⋂ n=1∞An=∅ . Prove that your example has the stated property.
An example is An on the interval (1, 1 + a/n] where n = 1, 2, ..., a and is
always a positive real number.
Proof by contradiction.
Through contradiction, suppose that ⋂
least one element. That element is x.

n=1



An is nonempty and has at

x is a member of the intersection, and because An is on the interval (1, 1
+ a/n], x > 1. Say that x-1 = ϵ.
As the limit of a/n is 0, there will always be a n0 that when m > n0, |a/m 0| < ϵ.
Therefore a/m < ϵ, because a is a positive real number. This can be
rewritten as a/m < x - 1, and then a/m + 1 < x.
Since x cannot fit in the interval (1, 1 + a/m], it cannot fit in the interval
(1, 1 + a/n]. X cannot exist, and the intersection ⋂ n=1∞An must be empty.


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