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- 07:55, 1 December 2020 Preimage of Subset under Mapping/Examples/Preimages of f(x, y) = x y (hist) [1,215 bytes] Prime.mover (talk | contribs) (Created page with "== Example of Preimage of Subset under Mapping == <onlyinclude> :File:Reciprocal-function.png|right|thumb|Yellow: $f^{-1} \sq...")
- 07:28, 1 December 2020 Continuous Mapping (Metric Space)/Examples/Identity Function with Discontinuity (hist) [1,677 bytes] Prime.mover (talk | contribs) (Created page with "== Example of Non-Continuous Mapping == <onlyinclude> Let $f: \R \to \R$ be the real function def...")
- 07:00, 1 December 2020 Continuous Mapping (Metric Space)/Examples (hist) [451 bytes] Prime.mover (talk | contribs) (Created page with "== Examples of Continuous Mappings (Metric Spaces) == <onlyinclude> </onlyinclude> Category:Examples of Continuous Mappings")
- 06:27, 1 December 2020 Preimage of Subset under Mapping/Examples/Identity Function with Discontinuity (hist) [1,585 bytes] Prime.mover (talk | contribs) (Created page with "== Example of Preimage of Subset under Mapping == <onlyinclude> Let $f: \R \to \R$ be the Definition:Real Function|real funct...")
- 23:07, 30 November 2020 Metric Space is Open in Itself/Examples/Closed Real Interval (hist) [999 bytes] Prime.mover (talk | contribs) (Created page with "== Examples of Use of Metric Space is Open in Itself == <onlyinclude> Let $\R$ be the real number line considered as an Definition:Euclid...")
- 23:03, 30 November 2020 Metric Space is Open in Itself/Examples (hist) [292 bytes] Prime.mover (talk | contribs) (Created page with "== Examples of Use of Metric Space is Open in Itself == <onlyinclude> </onlyinclude> Category:Metric Space is Open in Itself")
- 08:31, 30 November 2020 Open Ball of Point Inside Open Ball/Illustration (hist) [905 bytes] Prime.mover (talk | contribs) (Created page with "== Illustration of Open Ball of Point Inside Open Ball == <onlyinclude> This diagram illustrates the proof in $M = \struct {\R^2, d_2}$: :File:NeighborhoodInNeighborhoo...")
- 07:32, 30 November 2020 Open Ball of Metric Space/Examples/Real Number Line Example (hist) [1,420 bytes] Prime.mover (talk | contribs) (Created page with "== Example of Open Ball of Metric Space == <onlyinclude> Consider the real number line with the Defi...")
- 06:54, 30 November 2020 Open Ball of Metric Space/Examples (hist) [313 bytes] Prime.mover (talk | contribs) (Created page with "== Examples of Open Balls of Metric Spaces == <onlyinclude> </onlyinclude> Category:Examples of Open Balls")
- 22:57, 29 November 2020 Continuous Real-Valued Function/Examples/Non-Continuous Example (hist) [1,691 bytes] Prime.mover (talk | contribs) (Created page with "== Example of Continuous Real-Valued Function == <onlyinclude> Let $f: \R^2 \to \R$ be the Definition:Real Function of Two Var...")
- 22:37, 29 November 2020 Continuous Real-Valued Function/Examples (hist) [340 bytes] Prime.mover (talk | contribs) (Created page with "== Examples of Continuous Real-Valued Functions == <onlyinclude> </onlyinclude> Category:Continuous Real-Valued Functions")
- 18:13, 29 November 2020 Hilbert Cube is Compact/2-Sequence Space with 2-Norm (hist) [7,372 bytes] Julius (talk | contribs) (Created page with "== Theorem == <onlyinclude> Let $\ell^2$ be the 2-sequence space. Let $\norm {\, \cdot \,}_2$ be the 2-norm. Let $C$ be...")
- 22:02, 28 November 2020 Limit of Composite Function/Counterexample (hist) [1,240 bytes] Prime.mover (talk | contribs) (Created page with "== Counterexample to Limit of Composite Function is Image of Limit == <onlyinclude> Let $f$ and $g$ be the real functions defined as: {{begin-eqn...")
- 15:00, 28 November 2020 Convergence of Odd and Even Subsequences to Same Limit (hist) [2,592 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == Let $\sequence {s_n}$ be a real sequence. Let the subsequences $\sequence {s_{2 n} }$ and $\sequence {s...")
- 12:57, 28 November 2020 Limit of nth Root of Sum of nth Powers equals Maximum (hist) [1,308 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == Let $a_1, a_2, \ldots, a_r$ be non-negative real numbers. Let $n \in \Z_{>0}$ be a Definition:Strictly Positive Integ...")
- 11:35, 28 November 2020 Sum of r Powers is between Power of Maximum and r times Power of Maximum (hist) [1,004 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == Let $a_1, a_2, \ldots, a_r$ be non-negative real numbers. Let $n \in \Z_{>0}$ be a Definition:Strictly Positive Integ...")
- 11:35, 28 November 2020 Sum of r Powers is not Greater than r times Power of Maximum (hist) [1,186 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == <onlyinclude> Let $a_1, a_2, \ldots, a_r$ be non-negative real numbers. Let $n \in \Z_{>0}$ be a Definition:Strictly P...")
- 11:35, 28 November 2020 Power of Maximum is not Greater than Sum of Powers (hist) [973 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == <onlyinclude> Let $a_1, a_2, \ldots, a_r$ be non-negative real numbers. Let $n \in \Z_{>0}$ be a Definition:Strictly P...")
- 10:20, 28 November 2020 Min Operation on Continuous Real Functions is Continuous (hist) [1,785 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == <onlyinclude> Let $f: \R \to \R$ and $g: \R \to \R$ be real functions. Let $f$ and $g$ be Definition:Continuous Real Function at...")
- 10:20, 28 November 2020 Max Operation on Continuous Real Functions is Continuous (hist) [1,787 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == <onlyinclude> Let $f: \R \to \R$ and $g: \R \to \R$ be real functions. Let $f$ and $g$ be Definition:Continuous Real Function at...")
- 09:45, 28 November 2020 Absolute Value of Continuous Function is Continuous (hist) [1,583 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == <onlyinclude> Let $f: \R \to \R$ be a real function. Let $f$ be continuous at a po...")
- 13:11, 27 November 2020 Thomae Function is Continuous at Irrational Numbers (hist) [4,451 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == Let $D_M: \R \to \R$ denote the Thomae function: :$\forall x \in \R: \map {D_M} x = \begin {cases} 0 & : x = 0 \text { or } x \n...")
- 07:44, 27 November 2020 Dirichlet Function is Discontinuous (hist) [2,161 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == Let $D: \R \to \R$ denote the Dirichlet function: :$\forall x \in \R: \map D x = \begin {cases} c & : x \in \Q \\ d & : x \not...")
- 23:15, 26 November 2020 Min Operation Representation on Real Numbers (hist) [1,237 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == Let $x, y \in \R$. Then: :$\min \set{x, y} = \dfrac 1 2 \paren {x + y - \size {x - y} }$ where $\min$ denotes the min operation...")
- 23:08, 26 November 2020 Combination Theorem for Continuous Functions/Real/Difference Rule (hist) [1,099 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == Let $\R$ denote the real numbers. Let $f$ and $g$ be real functions which are Definition:Continuous...")
- 08:49, 25 November 2020 Set of Real Numbers is Bounded Below iff Set of Negatives is Bounded Above/Corollary (hist) [1,210 bytes] Prime.mover (talk | contribs) (Created page with "== Corollary to Set of Real Numbers is Bounded Below iff Set of Negatives is Bounded Above == <onlyinclude> Let $S$ be a subset of the Definition:R...")
- 08:29, 25 November 2020 Ordering of Real Numbers is Reversed by Negation (hist) [507 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == Let $x$ and $y$ in $\R$ be real numbers such that: :$x \le y$ Then: :$-y \le -x$ where $-y$ and $-x$ are the Definition:Negative o...")
- 07:49, 25 November 2020 Set of Real Numbers is Bounded Below iff Set of Negatives is Bounded Above (hist) [2,051 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == Let $S$ be a subset of the real numbers $\R$. Then: :$S$ is Definition:Bounded Below Subset of Real Numbers|...")
- 07:05, 25 November 2020 Continuum Property/Corollary (hist) [2,595 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == Let $S \subset \R$ be a non-empty subset of the set of real numbers such that $S$...")
- 23:23, 24 November 2020 Limit of Integer to Reciprocal Power/Proof 3 (hist) [884 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == {{:Limit of Integer to Reciprocal Power}} == Proof == <onlyinclude> {{ProofWanted|Follow the suggestion given in the source cited}} {{qed}} </onlyinclude> == S...")
- 23:16, 24 November 2020 Real Null Sequence/Examples/n^alpha over y^n (hist) [1,539 bytes] Prime.mover (talk | contribs) (Created page with "== Example of Real Null Sequence == <onlyinclude> Let $\alpha \in \R$ be a Definition:Strictly Positive Rational Number|(strictly) positive...")
- 22:56, 24 November 2020 Condition for Rational Number to be Square of Rational Number (hist) [775 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == Let $m$ and $n$ be (strictly) positive integers which are coprime. Then $\dfrac m n$ is...")
- 22:28, 24 November 2020 Supremum of Subset of Real Numbers/Examples/Example 8 (hist) [931 bytes] Prime.mover (talk | contribs) (Created page with "== Example of Supremum of Subset of Real Numbers == <onlyinclude> The subset $S$ of the Definition:Re...")
- 22:14, 24 November 2020 Supremum of Subset of Real Numbers/Examples/Example 6 (hist) [1,607 bytes] Prime.mover (talk | contribs) (Created page with "== Example of Supremum of Subset of Real Numbers == <onlyinclude> The subset $S$ of the Definition:Re...")
- 22:14, 24 November 2020 Supremum of Subset of Real Numbers/Examples/Example 7 (hist) [1,056 bytes] Prime.mover (talk | contribs) (Created page with "== Example of Supremum of Subset of Real Numbers == <onlyinclude> The subset $S$ of the Definition:Re...")
- 08:53, 24 November 2020 Supremum of Subset of Real Numbers/Examples/Example 5 (hist) [1,290 bytes] Prime.mover (talk | contribs) (Created page with "== Example of Supremum of Subset of Real Numbers == <onlyinclude> The subset $S$ of the Definition:Re...")
- 08:17, 24 November 2020 Union of Bounded Below Real Subsets is Bounded Below (hist) [1,521 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == Let $A$ and $B$ be sets of real numbers. Let $A$ and $B$ be Definition:Bounded Below Subset of Real Numbers|bou...")
- 08:16, 24 November 2020 Infimum of Union of Bounded Below Sets of Real Numbers (hist) [2,310 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == Let $A$ and $B$ be sets of real numbers. Let $A$ and $B$ both be Definition:Bounded Below Subset of Real Number...")
- 08:07, 24 November 2020 Infimum of Subset of Bounded Below Set of Real Numbers (hist) [1,414 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == Let $A$ and $B$ be sets of real numbers such that $A \subseteq B$. Let $B$ be Definition:Bounded Below Subset o...")
- 07:52, 24 November 2020 Subset of Bounded Below Set is Bounded Below (hist) [1,034 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == Let $A$ and $B$ be sets of real numbers such that $A \subseteq B$. Let $B$ be Definition:Bounded Below Subset o...")
- 07:52, 24 November 2020 Supremum of Union of Bounded Above Sets of Real Numbers (hist) [2,308 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == Let $A$ and $B$ be sets of real numbers. Let $A$ and $B$ both be Definition:Bounded Above Subset of Real Number...")
- 07:43, 24 November 2020 Union of Bounded Above Real Subsets is Bounded Above (hist) [1,526 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == Let $A$ and $B$ be sets of real numbers. Let $A$ and $B$ be Definition:Bounded Above Subset of Real Numbers|bou...")
- 23:34, 23 November 2020 Supremum of Subset of Bounded Above Set of Real Numbers (hist) [1,421 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == Let $A$ and $B$ be sets of real numbers such that $A \subseteq B$. Let $B$ be Definition:Bounded Above Subset o...")
- 23:24, 23 November 2020 Subset of Bounded Above Set is Bounded Above (hist) [1,007 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == Let $A$ and $B$ be sets of real numbers such that $A \subseteq B$. Let $B$ be Definition:Bounded Above Subset o...")
- 08:06, 23 November 2020 Combination Theorem for Sequences/Multiple Rule (hist) [939 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == === Real Sequences === {{:Combination Theorem for Sequences/Real/Multiple Rule}} === Combination The...")
- 05:15, 23 November 2020 Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Rank Axioms/Condition 2 Implies Condition 1 (hist) [2,230 bytes] Leigh.Samphier (talk | contribs) (Created page with "{{Proofread}} == Theorem == Let $S$ be a finite set. Let $\rho : \powerset S \to \Z$ be a mapping from the Definition:Pow...")
- 05:08, 23 November 2020 Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Rank Axioms/Condition 3 Implies Condition 2 (hist) [2,530 bytes] Leigh.Samphier (talk | contribs) (Created page with "{{Proofread}} == Theorem == Let $S$ be a finite set. Let $\rho$ is the rank function of a Definition:Matr...")
- 14:05, 22 November 2020 Combination Theorem for Limits of Functions/Complex/Quotient Rule (hist) [1,351 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == Let $\C$ denote the complex numbers. Let $f$ and $g$ be complex functions defined on an Defini...")
- 14:02, 22 November 2020 Combination Theorem for Limits of Functions/Complex/Product Rule (hist) [1,257 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == Let $\C$ denote the complex numbers. Let $f$ and $g$ be complex functions which are Definition...")
- 13:57, 22 November 2020 Combination Theorem for Limits of Functions/Complex/Combined Sum Rule (hist) [1,439 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == Let $\C$ denote the complex numbers. Let $f$ and $g$ be complex functions which are Definition...")