Closed Sets and Limits of Sequences

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Section Closed Sets and Limits of Sequences

Suppose we consider a sequence \((a_n)\) in a subset \(A\) of a metric space \(X\) that converges to a point \(x\text{.}\) Must it be the case that \(x \in A\text{?}\) We consider this question in the next activity.

Activity 10.11.

Let \(A = (0,1)\) and \(B = [0,1]\) in \((\R, d_E)\text{.}\) For each positive integer \(n\text{,}\) let \(a_n =\frac{1}{n}\text{.}\) Note that the sequence \((a_n)\) is contained in both sets \(A\) and \(B\text{.}\)

(a)

To what does the sequence \((a_n)\) converge in \(\R\text{?}\)

(b)

Is \(\lim a_n\) in \(A\text{?}\)

(c)

Is \(\lim a_n \in B\text{?}\)

(d)

Name two significant differences between the sets \(A\) and \(B\) that account for the different responses in parts (b) and \((c)\text{?}\) Respond using the terminology we have introduced in this section.

The result of Activity 10.11 is encapsulated in the next theorem.

Theorem 10.15.

A subset \(C\) of a metric space \(X\) is closed if and only if whenever \((c_n)\) is a sequence in \(C\) that converges to a point \(c \in X\text{,}\) then \(c \in C\text{.}\)

Proof.

Let \(X\) be a metric space and \(C\) a subset of \(X\text{.}\) First assume that \(C\) is closed. Let \((c_n)\) be a convergent sequence in \(C\) with \(c = \lim c_n\text{.}\) So either \(c \in C\) or \(c\) is a limit point of \(C\text{.}\) Since \(C\) contains its limit points, either case gives us \(c \in C\text{.}\) So \(\lim c_n \in C\text{.}\)

The proof of the remaining implication is left to the next activity.

Activity 10.12.

Let \(X\) be a metric space and \(C\) a subset of \(X\text{.}\) In this activity we will prove that if any time a sequence \((c_n)\) in \(C\) converges to a point \(c \in X\text{,}\) the point \(c\) is in \(C\text{,}\) then \(C\) is a closed set.

(a)

List three different ways that we can show that a subset of a metric space is closed. Which one might be relevant in this situation to show that the set \(C\) is closed?

(b)

Let \(c\) be a limit point of \(C\text{.}\) What does that tell us?

(c)

Complete the proof that \(C\) is a closed set.

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