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

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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This solution is comprised of a detailed explanation of the union of open intervals is also an open interval.
It contains step-by-step explanation for the following problem:

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 ).

Solution contains detailed step-by-step explanation.

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