Points where a sequence accumulates

Let $ (x_n) $ be a sequence of reals.

$ p \in \mathbb{R} $ is an accumulation point of the sequence if for every $ \epsilon > 0 $, $ x_n $ lies in $ (p-\epsilon, p+\epsilon) $ for infinitely many $ n $ (stricten "infinitely many" to "all but finitely many", and we get definition of limit of a sequence).

A subsequence of $ (x_n) $ is just a sequence $ x_{n_1}, x_{n_2}, \ldots $ with $ n_1 < n_2 < \ldots $

Notice $ \{$ limits of convergent subsequences of $ (x_n) \}$ = $ \{$ accumulation points of $ (x_n) \, \}$ :
A subsequential limit is clearly an accumulation point of seq. Say $ p \in \mathbb{R} $ is an accumulation point of seq. There is an $ n_1 $ with $ |x_{n_1} - p | < 1 $. Now as $ x_n $ lies in $ (p-\frac{1}{2}, p+\frac{1}{2}) $ for infinitely many $ n$, we can pick an $ n_2 > n_1 $ with $|x_{n_2} - p | < \frac{1}{2} $. And now we can pick an $ n_3 > n_2 $ with $|x_{n_3} - p | < \frac{1}{3} $, so on. This gives a subsequence $ (x_{n_k}) $ converging to $ p $.

Equivalence Relations

Let $ S $ be a set. Recall a partition of $ S $ is a collection of non-empty pairwise disjoint subsets with union $ S$.

Any partition $ P $ of $ S$ gives a relation $a \sim_P b \stackrel{\text{def}}{\iff} ( a, b \text{ are in same element of } P ) $.

One naturally wonders what $ \{ \sim_P \, : \, P \text{ a partition of } S \} $ is. Some thought reveals it is $ \{ \text{reflexive, symmetric, transitive relations on } S \} $.

Refl symm trans relations are usually called "equivalence relations".