## Definition of metric spaces

Definition. A metric space is a set $E$, together with a rule which associates with each pair $p, q \in E$ a real number $d \left( p, q \right)$ such that:

Proposition (Schwarz inequality):

Cauchy–Schwarz inequality in vector form:

Corollary of Schwarz inequality:

Or in vector form:

Proposition: generalize the triangle inequality to multi-angle inequality:

Proposition: difference of two sides of a triangle is less than the third side.

## Open and closed sets

Definition of open and closed balls:

Definition of open set:

Proposition: basic properties of metric spaces:

Proposition: an open ball is also an open set.

Definition of closed sets:

Proposition: a closed ball is also a closed set.

Definition of boundedness:

Proposition: a nonempty closed subset of $R$ contains its extrema.

## Convergence

Definition of convergence:

Uniqueness of convergence:

Definition of subsequence:

Subsequence of a convergent sequence is also convergent:

Convergent sequences are bounded:

Theorem. S is closed iff. convergence always occurs inside.

Proposition. If for two sequences $a_n, b_n$ with limits $a, b$ respectively, and $a_n \le b_n$ always holds, then $a \le b$.

Proposition. A bounded monotonic sequence of real numbers is convergent.

## Completeness

Definition of Cauchy sequence: elements are guaranteed to be arbitrarily close to each other (eventually).

Proposition, all convergent seqs in the metric space are Cauchy.

Because eventual closeness to the limit implies eventual closeness to each other.

Proposition: A Cauchy sequence that has a convergent subsequence is itself convergent.

Definition. A metric space is complete if every Cauchy sequence of points in E converges to a point in E.

Theorem: R is complete.

Theorem: For any positive integer n, $E^n$ is complete.

Definition of compactness:

Proposition. A compact subset of a metric space is bounded. In particular, a compact metric space is bounded.

Proposition: Nested set property.

Definition of cluster point.

Theorem: An infinite subset of a compact metric space has at least one cluster point.

## Math snippets

Compare $R^+$ with $A = \left\{ x^2 | x \in R^+ \text{ and } x \ne 0 \right\}$, note that $m \in R^+$ iff $m \in A$.

Cauchy–Schwarz inequality in vector form:
$\left| a \cdot b \right| \le \left| a \right| \left| b \right|$

We have two seqs: $a_n = 1/n, b_n = 1/n^2$, where $n = 2, 3, 4, 5, ...$. Obviously $a_n > b_n$, but

Proposition: generalize the triangle inequality to multi-angle inequality: