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].