Catalog of theorems from Real Analysis alongside proofs for each.

Lemma 4. Convergence via Norms
§4.2
Let (xk)⊆Rn. Thenlim⁡k→∞xk=a  ⟺  lim⁡k→∞∥xk−a∥=0.\text{Let } (\mathbf{x}_k) \subseteq \mathbb{R}^n. \text{ Then} \\ \lim_{k \to \infty} \mathbf{x}_k = \mathbf{a} \iff \lim_{k \to \infty} \lVert \mathbf{x}_k - \mathbf{a} \rVert = 0.

Lemma 5. Coordinatewise Convergence
§4.2
A sequence (xk)⊆Rn, xk=(xk,1⋮xk,n),converges to a=(a1,…,an) iffeach coordinate converges,i.e. lim⁡k→∞xk=a  ⟺  for each coordinate j=1,…,n, lim⁡k→∞xk,j=aj.\text{A sequence } (\mathbf{x}_k) \subseteq \mathbb{R}^n, \ \mathbf{x}_k = \begin{pmatrix} x_{k,1} \\ \vdots \\ x_{k,n} \end{pmatrix}, \\ \text{converges to $\mathbf{a} = (a_1, \ldots, a_n)$ iff} \\ \text{each coordinate converges,} \\ \text{i.e. } \lim_{k \to \infty} \mathbf{x}_k = \mathbf{a} \iff \\ \text{for each coordinate } j = 1, \ldots, n, \ \lim_{k \to \infty} x_{k,j} = a_j.

Lemma 6. Convergent Sequences are Cauchy
§4.2
Every convergent sequence is Cauchy.\text{Every convergent sequence is Cauchy.}

Theorem 7. Completeness Theorem for ℝⁿ
§4.2
Every Cauchy sequence in Rn converges.Thus, Rn is complete.\text{Every Cauchy sequence in $\mathbb{R}^n$ converges.} \\ \text{Thus, $\mathbb{R}^n$ is complete.}

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