Skip to the content

Math 2.0

General topology Proof

Proof that the closure is the set of limits of sequences

Builds on Metric space Closure Convergence of a sequence in a metric space Closure in a metric space The reals are Archimedean Closure is the smallest closed superset Closed sets contain their limits

Proof

We prove each direction separately.

Let $(X, d)$ be a metric space, let $S \subseteq X$ , let $x \in X$ , and let $\overline{S}$ be the closure of $S$ .
First let $x \in \overline{S}$ . By the description of the closure in a metric space, each $n \ge 1$ gives a point $s_n \in S$ with $d(x, s_n) < 1/n$ .
We show that $s_n \to x$ , in the sense of convergence. Let $\varepsilon > 0$ .
The reals are Archimedean, so there is a natural number $N$ with $N \varepsilon > 1$ . Then $N \ge 1$ , and $1/N < \varepsilon$ .
For $n \ge N$ , symmetry of $d$ gives $d(s_n, x) = d(x, s_n) < 1/n \le 1/N < \varepsilon$ . So $s_n \to x$ .
Conversely, let $(s_n)$ be a sequence of points of $S$ with $s_n \to x$ .
By closure is the smallest closed superset, $S \subseteq \overline{S}$ and $\overline{S}$ is closed. So $(s_n)$ is a sequence of points of the closed set $\overline{S}$ .
Since closed sets contain their limits, we get $x \in \overline{S}$ .

Idea

A point of the closure has points of $S$ at distance less than $1/n$ for every $n$ , and these form a sequence that tends to it. Conversely, the closure is a closed set that contains $S$ . So it keeps every limit of points of $S$ .

Reading path

To understand this page from the ground up, read these 86 pages in order. The foundations come first. The path takes the proofs Proof of the approximation property of the supremum, Proof of De Morgan's laws for sets, Proof of De Morgan's laws by truth tables, Proof that a union of open sets of a metric space is open, Proof that an intersection of closed sets of a metric space is closed, Proof that the reals form an ordered field, with the Cauchy sequence construction, Proof that the closure is the smallest closed superset, Proof of the description of the open sets of a metric space, Proof of the description of the closure in a metric space, Proof that closed sets contain their limits, Proof that the reals are Archimedean from the completeness axiom, Proof that open balls are open, Proof that the closure is closed.

Construction of the reals from Cauchy sequences concept
Element definition
Integers object
Ordered field definition
Proposition concept
Rational numbers object
Real numbers object
Set bounded above definition
Set definition
Ring of integers statement
Archimedean ordered field definition
Conjunction definition
Disjunction definition
Implication definition
Negation definition
Predicate definition
The reals form a field statement
Supremum definition
Empty set object
Family of sets definition
Ordered pair definition
Intersection definition
Union definition
The rationals form a field statement
Biconditional definition
Truth table concept
Existential quantifier definition
Universal quantifier definition
Completeness axiom statement
Cartesian product definition
Subset definition
Axiom of extensionality statement
The rationals form an ordered field statement
Intersection of a family of sets definition
Union of a family of sets definition
Propositional logic concept
Uniqueness quantifier definition
Binary relation definition
Universal set definition
Order on the reals definition
Topological space definition
Tautology definition
Antisymmetric relation definition
Reflexive relation definition
Transitive relation definition
Complement definition
Function definition
Logical equivalence definition
Preorder definition
Metric space definition
Proof of De Morgan's laws by truth tables proof
Partial order definition
Convergence of a sequence in a metric space definition
Open ball definition
Triangle inequality for a metric statement
De Morgan's laws statement
Total order definition
Metric topology definition
Proof of De Morgan's laws for sets proof
Open set definition
Proof that the reals form an ordered field, with the Cauchy sequence construction proof
De Morgan's laws for sets statement
Proof of the description of the open sets of a metric space proof
Closed set definition
The reals form an ordered field statement
Open sets of a metric space statement
Closure definition
Proof of the approximation property of the supremum proof
Proof that closed sets contain their limits proof
Proof that open balls are open proof
Proof that a union of open sets of a metric space is open proof
Proof that the closure is closed proof
Approximation property of the supremum statement
Closed sets contain their limits statement
Open balls are open statement
Union of open sets is open statement
Proof that the reals are Archimedean from the completeness axiom proof
Proof of the description of the closure in a metric space proof
Proof that an intersection of closed sets of a metric space is closed proof
The reals are Archimedean statement
Closure in a metric space statement
Intersection of closed sets is closed statement
Closure is closed statement
Proof that the closure is the smallest closed superset proof
Closure is the smallest closed superset statement
Proof that the closure is the set of limits of sequences proof

Where this leads

5 pages build on this one.

Further reading

Web https://en.wikipedia.org/wiki/Closure_(topology)

About this page

Tag	`ZF18TYQMSZ`
URI	`m2://ZF18TYQMSZ`
Kind	proof
Writing	W0 stub
MSC	54E35
Proves	Closure is the set of limits of sequences
Uses	Metric space, Closure, Convergence of a sequence in a metric space, Closure in a metric space, The reals are Archimedean, Closure is the smallest closed superset, Closed sets contain their limits
Changed	4 November 2026

Markdown · JSON · Edit on GitHub

What this page needs for W1

an inventory row: the row of the inventory that the node accounts for (0 of 1)
a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.