It is most efficient to discus things in terms of abstract metric spaces to begin with.
Definition 2.1.1 A non empty set X is called a metric space if there is a function d : X × X → [0,∞) which satisfies the following axioms.
This function d is called the metric. We often refer to it as the distance also.
Definition 2.1.2 An open ball, denoted as B

A set U is said to be open if whenever x ∈ U, it follows that there is r > 0 such that B
For example, you could have X be a subset of ℝ and d
Then the first thing to show is the following.
Proof: Suppose y ∈ B

Thus y ∈ B
Definition 2.1.4 Let S be a nonempty subset of a metric space. Then p is a limit point (accumulation point) of S if for every r > 0 there exists a point different than p in B
A related idea is the notion of the limit of a sequence. Recall that a sequence is really just a mapping from ℕ to X. We write them as
Definition 2.1.5 We say that x = lim_{n→∞}x_{n} when for every ε > 0 there exists N such that if n ≥ N, then

Often we write x_{n} → x for short. This is equivalent to saying

Proposition 2.1.6 The limit is well defined. That is, if x,x^{′} are both limits of a sequence, then x = x^{′}.
Proof: From the definition, there exist N,N^{′} such that if n ≥ N, then d

Since ε is arbitrary, this shows that x = x^{′} because d
Next there is an important theorem about limit points and convergent sequences.
Theorem 2.1.7 Let S≠∅. Then p is a limit point of S if and only if there exists a sequence of distinct points of S,
Proof:
⇐= Conversely, if such a sequence
Note that this says that the complement of an open set is closed. If V is open, then the complement of its complement is itself. Thus
Then the following theorem gives the relationship between closed sets and limit points.
Proof:
⇐= Next suppose H has all of its limit points. Why is H^{C} open? If p ∈ H^{C} then it is not a limit point and so there exists δ > 0 such that B
Corollary 2.1.10 A set H is closed if and only if whenever
Proof:
⇐= Suppose the limit condition holds, why is H closed? Let x ∈ H^{′} the set of limit points of H. By Theorem 2.1.7 there exists a sequence of points of H,
Next is the important concept of a subsequence.
Definition 2.1.11 Let
The really important thing about subsequences is that they preserve convergence.
Theorem 2.1.12 Let

also.
Proof: Let ε > 0 be given. Then there exists N such that

It follows that if k ≥ N, then n_{k} ≥ N and so

This is what it means to say lim_{k→∞}x_{nk} = x. ■