# Sequences

## Sequences

• Notation
• $$\{x_1, x_2, ... \}$$
• $$\{x_i\}_{i=1}^\infty$$
• $$\{x_n\}$$
• Different from sets in that repetitions and order matter

## Metric space

A metric space is a set, $$X$$, and function $$d:X\times X \to \mathbb{R}$$ such that $$\forall x, y, z \in X$$

1. $$d(x,y) > 0$$ unless $$x=y$$ and then $$d(x,x) = 0$$
2. (symmetry) $$d(x,y) = d(y,x)$$
3. (triangle inequality) $$d(x,y) \leq d(x,z) + d(z,y)$$.

## Metric space examples

$$\mathbb{R}$$ with $$d(x,y) = |x-y|$$

## Metric space examples

• $$\mathbb{R}^n$$ with $$d(x,y) = \sqrt{\sum_{i=1}^n (x_i - y_i)^2}$$
• $$\mathbb{R}^n$$ with $$d_p(x,y) = \left(\sum_{i=1}^n \vert x_i - y_i \vert^p \right)^{1/p}$$ with $$p \geq 1$$

## Convergence

• $$\{x_n\}_{n=1}^\infty$$ in a metric space converges to $$x$$ if $$\forall\epsilon>0\, \exists N$$ such that $$d(x_n,x)<\epsilon$$ for all $$n \geq N$$
• $$x$$ is the limit of $$\{x_n\}_{n=1}^\infty$$
• written $$\lim_{n \rightarrow \infty} x_n = x$$ or $$x_n \rightarrow x$$.

## Accumulation points

• $$a$$ is an accumulation point of $$\{x_n\}_{n=1}^\infty$$ if $$\forall \epsilon > 0$$ $$\exists$$ infinitely many $$x_i$$ such that $$d(a, x_i ) < \epsilon$$
• Lemma: If $$x_n \rightarrow x$$, then $$x$$ is the only accumulation point of $$\{x_n\}_{n=1}^\infty$$

## Subsequence

• Given $$\{x_n\}_{n=1}^\infty$$ and any sequence of positive integers, $$\{n_k\}$$ such that $$n_1 < n_2 <$$ … we call $$\{x_{n_k}\}_{k=1}^\infty$$ a subsequence of $$\{x_n\}_{n=1}^\infty$$
• Lemma: Let $$a$$ be an accumulation point of $$\{x_n\}$$, then $$\exists$$ a subsequence that converges to $$a$$.

## Cauchy sequence

• $$\{x_n\}_{n=1}^\infty$$ is a Cauchy sequence if for any $$\epsilon > 0$$ $$\exists N$$ such that for all $$i,j\geq N$$, $$d(x_i,x_j) < \epsilon$$
• Metric space, $$X$$, is complete if every Cauchy sequence of points in $$X$$ converges in $$X$$
• $$\mathbb{R}$$ is defined as a the unique complete ordered field

## Metric space examples

• $$\mathbb{R}^n$$ is complete

## Metric space examples

• $$\ell_p = \{ (x_1, x_2, ...)\, s.t.\, x_i \in \mathbb{R}, \sum_{i=1}^\infty |x_i|^p < \infty \}$$ with metric $d_p(x,y) = \left( \sum_{i=1}^\infty |x_i - y_i|^p \right)^{1/p}$

• $$\ell_\infty = \{ (x_1, x_2, ...)\, s.t.\, x_i \in \mathbb{R}, \sup_i |x_i| < \infty \}$$ with metric $d_\infty(x,y) = \sup_i |x_i - y_i|$

• $$\ell_p$$ is complete

# Topology

## Open sets

• A neighborhood of $$x$$ is the set $N_\epsilon (x) = \{y \in X: d(x,y) < \epsilon \}$

• $$S \subseteq X$$ is open if $$\forall x \in S$$, $$\exists$$ $$\epsilon>0$$ such that $N_\epsilon(x) \subset S$

## Opens sets

• Theorem:
• Any union of open sets is open (finite or infinite)
• The finite intersection of open sets is open

## Closed sets

• A set $$S \subseteq X$$ is closed if its complement, $$S^c$$, is open

## Closed sets

• Theorem:
• The intersection of any collection of closed sets is closed
• The union of any finite collection of closed sets is closed

## Convergence

• $$x_n \to x$$ if and only if for every open set $$U$$ containing $$x$$ $$\exists N$$ such that $$x_n \in U$$ for all $$n \geq N$$
• Let $$\{x_n\}$$ be any convergent sequence with each element in $$C$$, then $$\lim x_n = x \in C$$ for all such $$\{x_n\}$$ if and only if $$C$$ is closed

# Compactness

## Compact sets

• An open cover of a set $$S$$ is a collection of open sets, $$\{G_\alpha\}$$ $$\alpha \in \mathcal{A}$$ such that $$S \subset \cup_{\alpha \in \mathcal{A}} G_\alpha$$
• A set $$K$$ is compact if every open cover of $$K$$ has a finite subcover

## Compact sets

• Lemma If $$K$$ is compact, then $$K$$ is closed
• Lemma If $$K$$ is compact, then $$K$$ is bounded
• $$K$$ is bounded if $$\exists x_0 \in K$$ and $$r \in \mathbb{R}$$ such that $d(x,x_0) < r$ for all $$x \in K$$

## Heine-Borel

• Theorem: A set $$S \subseteq \mathbb{R}^n$$ is compact if and only if it is closed and bounded

## Sequential compactness

• $$K$$ is sequentially compact if every sequence in $$K$$ has an accumulation point in $$K$$
• Lemma: If $$K$$ is compact, then $$K$$ is sequentially compact
• Theorem: Let $$X$$ be a metric space and $$K \subseteq X$$. $$K$$ is compact if and only if $$K$$ is sequentially compact

# Functions and continuity

## Functions

• Function from a set $$A$$ to a set $$B$$ is a rule that assigns to each $$a \in A$$ one and only one $$b \in B$$
• Domain $$A$$
• Target space $$B$$
• Range or image $$\{y \in B: f(x) = y \text{ for some } x \in A \}$$

## Continuity

• $$f:X \to Y$$ is continuous at $$x$$ if whenever $$x_n \to x$$ in $$X$$, then $$f(x_n) \to f(x)$$ in $$Y$$.
• $$f: X \to Y$$ is continuous at $$x$$ if and only if for every $$\epsilon>0$$ $$\exists$$ $$\delta>0$$ such that $$d(x,x') < \delta$$ implies $$d(f(x),f(x')) < \epsilon$$
• $$f:X \to Y$$ is continuous at $$x$$ if and only if for all open $$V$$ with $$f(x) \in V$$, $$\exists$$ open $$U \subseteq X$$ such that $$x \in U \subseteq f^{-1}(V)$$
• Preimage $$f^{-1} (V) = \{ x \in X: f(x) \in V \}$$

## Existence of maximum

• Let $$f:X \to \mathbb{R}$$ be continuous and $$K \subset X$$ be compact. Then $$\exists x^* \in K$$ such that $$f(x^*) \geq f(x) \forall x \in K$$