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    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.

    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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