📐 Theorem 2.47 (Connected Subsets of ℝ) A subset $E$ of $\\mathbb{R}$ is connected if and only if: whenever $x, y \\in E$ and $x < z < y$, we have $z \\in E$. That is, the connected subsets of $\\mathbb{R}$ are precisely the intervals. From: rudin Learn more:

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📐 Theorem 10.5.1 (L Let $f, g : (a, b) \\to \\mathbb{R}$ be differentiable with $g\ From: tao-analysis-1 Learn more:

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📖 Euler $\\phi(n)$ is the number of integers from 1 to $n$ that are relatively prime to $n$. From: intro-discrete Learn more:

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📖 Definition 5.5.1 (Upper Bound) Let $E$ be a subset of $\\mathbb{R}$, and let $M$ be a real number. We say $M$ is an **upper bound** for $E$ iff $x \\leq M$ for every $x \\in E$. From: tao-analysis-1 Learn more:

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📐 Distance Formula The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is $d = \\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$ Proof: By the Pythagorean theorem applied to the right triangle formed by the horizontal and vertical displacements: $d^2 = (x_2-x_1)^2 + (y_2-y_1)^2$ Therefore $d = \\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$ From: Men of Mathematics Learn more:

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An interactive journey through 2500 years of mathematical history with E.T. Bell

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📖 Definition 4.3.4 (ε-Closeness for Rationals) Let $\\varepsilon > 0$ be a rational. We say rationals $x$ and $y$ are **$\\varepsilon$-close** iff $|x - y| \\leq \\varepsilon$. From: tao-analysis-1 Learn more:

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📖 Exponential Distribution A random variable $X$ has an \\textbf{exponential distribution} with parameter $\\lambda > 0$ if its distribution function is $F(t) = 1 - e^{-\\lambda t}$ for $t \\geq 0$, and $F(t) = 0$ for $t < 0$. The density function is $f(t) = \\lambda e^{-\\lambda t}$ for $t \\geq 0$. From: calc2 Learn more:

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💡 Proposition III.12 If two circles touch one another externally, the straight line joining their centres will pass through the point of contact. From: Euclid's Elements Learn more:

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The foundational text of Western mathematics. Study all 13 books of Euclid

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💡 The Principle of Correlation Between Planes Every being, force, and form on the physical plane has its counterpart on the astral and devachanic planes. Physical matter is the condensed expression of astral forces, which are in turn the densified expression of devachanic archetypes. What appears as a physical object to the senses appears as a living, luminous form on the astral plane and as a sounding, creative thought-being on the devach... From: steiner-GA90a Learn more:

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📐 Theorem 4.14 (Image of Compact) If $f: X \\to Y$ is continuous and $K \\subset X$ is compact, then $f(K)$ is compact. Proof: Let $\\{V_\\alpha\\}$ be an open cover of $f(K)$. Since $f$ is continuous, each $f^{-1}(V_\\alpha)$ is open. These sets cover $K$: $K \\subset \\bigcup_\\alpha f^{-1}(V_\\alpha)$. Since $K$ is compact, finitely many suffice: $K \\subset f^{-1}(V_{\\alpha_1}) \\cup \\cdots \\cup f^{-1}(V_{\\alp... From: rudin Learn more:

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📖 Ends and Means Every action aims at an end (goal) and employs means to achieve it. Means are always scarce relative to ends; otherwise no action would be needed. From: Man, Economy, and State Learn more:

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Murray Rothbard

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💡 Hidden Purpose and Sacrifice The apparent abandonment of great potential by Francisco and other strikers is actually purposeful sacrifice in service of a larger cause. From: Atlas Shrugged Learn more:

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📖 Invertible Matrix An $n \\times n$ matrix $A$ is \\textbf{invertible} (or \\textbf{nonsingular}) if there exists an $n \\times n$ matrix $A^{-1}$ such that $AA^{-1} = A^{-1}A = I_n$. A matrix that is not invertible is called \\textbf{singular}. From: calc2 Learn more:

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📐 Theorem 11.41 (Riesz-Fischer) $L^2(\\mu)$ is complete with respect to the norm $\\|f\\|_2 = \\left(\\int |f|^2\\right)^{1/2}$. From: rudin Learn more:

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📖 Linear Transformation A linear transformation $T: \\mathbb{R}^n \\to \\mathbb{R}^m$ satisfies $T(\\mathbf{u} + \\mathbf{v}) = T(\\mathbf{u}) + T(\\mathbf{v})$ and $T(c\\mathbf{v}) = cT(\\mathbf{v})$. From: Four Pillars of Geometry Learn more:

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💡 Equilateral Triangle Construction (Euclid I.1) Given a line segment $AB$, construct circles centered at $A$ and $B$ with radius $|AB|$. If they intersect at point $C$, then $\\triangle ABC$ is equilateral. Proof: Since $C$ lies on the circle centered at $A$, we have $|AC| = |AB|$ by definition of circle. Since $C$ lies on the circle centered at $B$, we have $|BC| = |AB|$ by definition of circle. Therefore $|AB| = |AC| = |BC|$, making $\\triangle ABC$ equilateral. From: Four Pillars of Geometry Learn more:

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📐 Euclidean Algorithm $\\gcd(a,b) = \\gcd(b, a \\mod b)$. Repeated application computes $\\gcd(a,b)$ in $O((\\log b)^2)$ bit operations. Proof: If $a = bq + r$, then any common divisor of $a$ and $b$ also divides $r = a - bq$. Conversely, any common divisor of $b$ and $r$ also divides $a = bq + r$. Therefore $\\gcd(a,b) = \\gcd(b,r)$. The algorithm terminates because remainders strictly decrease. Each step halves the larger numbe... From: Algebraic Number Theory Learn more:

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📖 Injective Module An $R$-module $Q$ is \\textbf{injective} if for every injection $f: M \\to N$ and map $g: M \\to Q$, there exists an extension $\\tilde{g}: N \\to Q$ with $\\tilde{g} \\circ f = g$. From: df-course Learn more:

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📐 Absolute Convergence Implies Convergence If $\\sum |a_n|$ converges, then $\\sum a_n$ converges. From: Real Analysis Learn more:

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📐 Pappus Let $A, B, C$ be collinear and $D, E, F$ be collinear on a parallel line. If $AB \\parallel ED$ and $FE \\parallel BC$, then $AF \\parallel CD$. Proof: By similar triangles formed by the parallel lines, we establish proportions that force $AF \\parallel CD$. From: Four Pillars of Geometry Learn more:

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