Prove that a closed interval \([c,d] = \{c \leq x \leq d\}\) is a closed set.


Proof

Let \([c,d] = \{c \leq x \leq d\}\) be a closed interval. To prove that it’s a closed set, we need to prove that \([c,d]\) contains all of its limit points so let \(x\) be an arbitrary limit point. By the definition of a limit point, we know that there must exist a sequence \((x_n)\) such that \((x_n) \rightarrow x\). We need to show that \(x \in [c,d]\). By the order limit theorem (iii), since \(c \leq x_n \leq d\) for all \(n \in \mathbf{N}\) it follows that \(c \leq x \leq d\) as well. Therefore, \([c,d]\) is closed as required. \(\blacksquare\).

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