Thread: proving a set is closed

how can I prove that [a, b] is a closed set?

A set is closed if its complement is open.

In this case prove:
(-inf, a) union (b, +inf)
is open.

To prove this set is open just use the definition.
Intuitively: show for every point in the open set, there is a small neighborhood around that point that is also in the set.

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You can also use the lemma that a set is closed iff the closure is the set itself. I.e. show the closure of [a,b] is just [a,b].

Originally Posted by cottekr

how can I prove that [a, b] is a closed set?

It depends what your definition of closed is!

Originally Posted by cottekr

how can I prove that [a, b] is a closed set?

To prove that [a,b] is closed you must prove:

1) For all x>b there exists an ε>0 ,such that:

......For all ,y : $\displaystyle |x-y|<\epsilon\Longrightarrow y>b$.

2) For all x<a there exists an ε>0 , such that:

......For all ,y : $\displaystyle |x-y|<\epsilon\Longrightarrow y<a$