Catalog of Definitions from Real Analysis.

Definition 9. Convergence of a Sequence
§4.2
A sequence of points (xk)⊆Rnconverges to a point a∈Rn if∀ε>0, ∃N∈N, ∀k≥N, ∥xk−a∥<ε.In this case, we notate this by lim⁡k→∞xk=a.\text{A sequence of points } (\mathbf{x}_k) \subseteq \mathbb{R}^n \\ \text{converges to a point } \mathbf{a} \in \mathbb{R}^n \text{ if} \\ \forall \varepsilon > 0,\ \exists N \in \mathbb{N},\ \forall k \geq N,\ \lVert \mathbf{x}_k - \mathbf{a} \rVert < \varepsilon. \\ \text{In this case, we notate this by } \lim_{k \to \infty} \mathbf{x}_k = \mathbf{a}.
Definition 10. Cauchy Sequence
§4.2
A sequence (xk)⊆Rn is Cauchy if∀ε>0, ∃N∈N, ∀k,l≥N, ∥xk−xl∥<ε.\text{A sequence } (\mathbf{x}_k) \subseteq \mathbb{R}^n \text{ is Cauchy if} \\ \forall \varepsilon > 0,\ \exists N \in \mathbb{N},\ \forall k, l \geq N,\ \lVert \mathbf{x}_k - \mathbf{x}_l \rVert < \varepsilon.
Definition 11. Complete
§4.2
A set S⊆Rn is complete if everyCauchy sequence‾ in S converges‾ to a point in S.(xk)⊆S, (xk) Cauchy  ⟹  ∃a∈Rn, lim⁡k→∞xk=a∈S\text{A set } S \subseteq \mathbb{R}^n \text{ is complete if every} \\ \underline{\href{/notes/real-analysis/definitions#definition-10}{\text{Cauchy sequence}}} \text{ in } S \text{ } \underline{\href{/notes/real-analysis/definitions#definition-9}{\text{converges}}} \text{ to a point in } S. \\[0.5em] (\mathbf{x}_k) \subseteq S,\ (\mathbf{x}_k) \text{ Cauchy} \implies \exists \mathbf{a} \in \mathbb{R}^n,\ \lim_{k \to \infty} \mathbf{x}_k = \mathbf{a} \in S
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