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

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

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

• 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 \(f(x^*) \geq f(x) \forall x \in K\)