- Notation
- \(\{x_1, x_2, ... \}\)
- \(\{x_i\}_{i=1}^\infty\)
- \(\{x_n\}\)

- Different from sets in that repetitions and order matter

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

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

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

- \(\mathbb{R}^n\) 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 \(p \geq 1\)

- \(\{x_n\}_{n=1}^\infty\) 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 \(x_n \rightarrow x\).

- \(a\) 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 \(\{x_n\}_{n=1}^\infty\)

- Given \(\{x_n\}_{n=1}^\infty\) and any sequence of positive integers, \(\{n_k\}\) such that \(n_1 < n_2 <\) â€¦ 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 \(a\).

- \(\{x_n\}_{n=1}^\infty\) 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

- \(\mathbb{R}^n\) is complete

\(\ell_p = \{ (x_1, x_2, ...)\, s.t.\, x_i \in \mathbb{R}, \sum_{i=1}^\infty |x_i|^p < \infty \}\) 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

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 \[ N_\epsilon(x) \subset S \]

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

- A set \(S \subseteq X\) is
**closed**if its complement, \(S^c\), is open

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

- \(x_n \to x\) 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

- 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

**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 \(x \in K\)

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

- \(K\) 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

- 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 \(\{y \in B: f(x) = y \text{ for some } x \in A \}\)

- \(f:X \to Y\) 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 \(f^{-1} (V) = \{ x \in X: f(x) \in V \}\)

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