Hi, I am having problems with proving the double containment argument for the following problems. I would appreciate any feedback. Thanks.(adsbygoogle = window.adsbygoogle || []).push({});

Compute [tex]\bigcup[/tex] (i [tex]\in[/tex] I) Ui and [tex]\bigcap[/tex] (i [tex]\in[/tex] I) Ui if

a. I=N, the set of all positive natural numbers and U_n = (-n,n) [tex]\subseteq[/tex]R(U_n is the open interval between -n and n)

i.

[tex]\bigcap[/tex] (i [tex]\in[/tex] I) Ui = (infinity)

How do I prove the double containment argument?

Do I say that if x is not equal to infinity then there exists a number such that x=1?

ii.

[tex]\bigcup[/tex] (i [tex]\in[/tex] I) Ui = (-1,1)

Do I write is x is a member of (-1,1) then there exists a number such that |x| < 1 or 1< 1/|x|

b. I=N, the set of all real numbers and U_r = (0, 1/(1+r^2)) [tex]\subseteq[/tex]R(U_r is the open interval between 0 and 1/(1+r^2))

i.

[tex]\bigcap[/tex] (i [tex]\in[/tex] I) Ui = (0,0)

For the double containment argument do I write what if x is not equal to 0?

ii.

[tex]\bigcup[/tex] (i [tex]\in[/tex] I) Ui = (0, 0.5)

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# Infinite Operations with Sets

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