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].
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 : $\displaystyle |x-y|<\epsilon\Longrightarrow y<a$