proof9 .pdf

File information


Original filename: proof9.pdf
Author: vargh

This PDF 1.7 document has been generated by WPS Office / , and has been sent on pdf-archive.com on 28/08/2017 at 01:39, from IP address 108.218.x.x. The current document download page has been viewed 185 times.
File size: 74 KB (1 page).
Privacy: public file


Download original PDF file


proof9.pdf (PDF, 74 KB)


Share on social networks



Link to this file download page



Document preview


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.


Document preview proof9.pdf - page 1/1


Related documents


proof9
proof10
math words
graph theory and applications rev4
chas prof chiriano english
odelaclahane061416

Link to this page


Permanent link

Use the permanent link to the download page to share your document on Facebook, Twitter, LinkedIn, or directly with a contact by e-Mail, Messenger, Whatsapp, Line..

Short link

Use the short link to share your document on Twitter or by text message (SMS)

HTML Code

Copy the following HTML code to share your document on a Website or Blog

QR Code

QR Code link to PDF file proof9.pdf