# Prove that union I: = Uα Iα is an open interval.

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Fix a point p in R. Let { Iα } be a ( possibly infinite ) collection of open intervals Iα = ( cα , dα )

which is a subset of R, such that pЄ Iα for all α.

Prove that the union I: = Uα Iα is also an open interval ( possibly infinite ).

Hint: Consider c: = infα cα and d: = supα dα and show that I = ( c, d ).

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#### Solution Preview

Please see the attached file.

Let

Since union of two open intervals which contain p is an open interval,

where smaller of and

and greater of and .

Again ,

...

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