Catalog of theorems from Real Analysis alongside proofs for each.

Theorem 1. Cauchy-Schwarz Inequality
§4.1
For x,y∈Rn, ∣⟨x,y⟩∣≤∥x∥∥y∥Equality holds iff x and y are collinear.\text{For } \mathbf{x}, \mathbf{y} \in \mathbb{R}^n, \ |\langle x, y \rangle | \leq \| \mathbf{x} \| \| \mathbf{y} \| \\ \text{Equality holds iff } \mathbf{x} \text{ and } \mathbf{y} \text{ are collinear}.

Lemma 2. Properties of Orthonormal Sets
§4.1
Let {v1,…,vn}⊆Rn be an orthonormal set.(a) For aj∈R, ∥∑j=1najvj∥=(∑j=1naj2)1/2(b) Any orthonormal set in Rn is linearly independent.(c) An ONB in Rn is a basis and has exactly n elements.\text{Let } \{\mathbf{v}_1, \ldots, \mathbf{v}_n\} \subseteq \mathbb{R}^n \text{ be an orthonormal set.} \\[0.5em] \text{(a) For } a_j \in \mathbb{R},\ \left\lVert \sum_{j=1}^{n} a_j \mathbf{v}_j \right\rVert = \left( \sum_{j=1}^{n} a_j^2 \right)^{1/2} \\[0.5em] \text{(b) Any orthonormal set in } \mathbb{R}^n \text{ is linearly independent.} \\[0.5em] \text{(c) An ONB in } \mathbb{R}^n \text{ is a basis and has exactly } n \text{ elements.}

Lemma 3. Triangle Inequality
§4.1
For all x,y∈Rn, ∥x+y∥≤∥x∥+∥y∥.Moreover, the equality holds iffx=0 or ∃c≥0, y=cx.\text{For all } \mathbf{x}, \mathbf{y} \in \mathbb{R}^n,\ \lVert \mathbf{x} + \mathbf{y} \rVert \leq \lVert \mathbf{x} \rVert + \lVert \mathbf{y} \rVert. \\ \text{Moreover, the equality holds iff} \\ \mathbf{x} = \mathbf{0} \text{ or } \exists c \geq 0,\ \mathbf{y} = c\mathbf{x}.

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.}

Proposition 8. Unions and Intersections of Closed Sets
§4.3
If A,B⊂Rn are closed, then A∪B is closed.If {Ai:i∈I} is a family of closed subsets of Rn,then ⋂i∈IAi is closed.\text{If } A, B \subset \mathbb{R}^n \text{ are closed, then } A \cup B \text{ is closed.} \\ \text{If } \{A_i : i \in I\} \text{ is a family of closed subsets of } \mathbb{R}^n, \\ \text{then } \bigcap_{i \in I} A_i \text{ is closed.}

Lemma 9. Closure is the Smallest Closed Set
§4.3
Let A⊆Rn. Then A‾ is the smallest closed set containing A.In particular, A‾‾=A‾.\text{Let } A \subseteq \mathbb{R}^n. \text{ Then } \overline{A} \text{ is the smallest closed set containing } A. \\ \text{In particular, } \overline{\overline{A}} = \overline{A}.

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