hcju/proofwikiqa huggingface dataset:
{{begin-eqn}} {{eqn | l = \map {\dfrac \d {\d t} } {\map F t \map G t}       | r = \map f t \map G t + \map F t \map g t       | c = Product Rule for Derivatives }} {{eqn | ll= \leadsto       | l = \int \paren {\map f t \map G t + \map F t \map g t} \rd t       | r = \map F t \map G t       | c = Fundamental Theorem of Calculus: integrating both sides {{WRT|Integration}} $t$ }} {{eqn | ll= \leadsto       | l = \int \map f t \map G t \rd t + \int \map F t \map g t \rd t       | r = \map F t \map G t       | c = Linear Combination of Primitives }} {{eqn | ll= \leadsto       | l = \int \map f t \map G t \rd t       | r = \map F t \map G t - \int \map F t \map g t \rd t       | c = rearranging }} {{end-eqn}} {{qed}}
Let $F$ be an antiderivative of $f$. We have: {{begin-eqn}} {{eqn | l = \map {\frac \d {\d u} } {\map F x}       | r = \map {\frac \d {\d u} } {\map F {\map \phi u} }       | c = Definition of $\map \phi u$ }} {{eqn | r = \dfrac \d {\d x} \map F {\map \phi u} \dfrac \d {\d u} \map \phi u       | c = Chain Rule for Derivatives }} {{eqn | r = f {\map \phi u} \dfrac \d {\d u} \map \phi u       | c = as $\map F x = \ds \int \map f x \rd x$ }} {{end-eqn}} Hence $\map F {\map \phi u}$ is an antiderivative of $\map f {\map \phi u} \d {\d u} \map \phi u$. Thus: {{begin-eqn}} {{eqn | l = \int_a^b \map f {\map \phi u} \map {\phi'} u \rd u       | r = \bigintlimits {\map F {\map \phi u} } a b       | c = Fundamental Theorem of Calculus: Second Part }} {{eqn | n = 1       | r = \map F {\map \phi b} - \map F {\map \phi a}       | c =  }} {{end-eqn}} However, also: {{begin-eqn}} {{eqn | l = \int_{\map \phi a}^{\map \phi b} \map f t \rd t       | r = \bigintlimits {\map F t} {\map \phi a} {\map \phi b}       | c =  }} {{eqn | r = \map F {\map \phi b} - \map F {\map \phi a}       | c =  }} {{eqn | r = \int_a^b \map f {\map \phi u} \map {\phi'} u \rd u       | c = from $(1)$ }} {{end-eqn}} which was to be proved.  {{qed}}
=== Case 1 : $z \in \Z_{>0}$ === Let $z \in \Z_{>0}$. From Fundamental Theorem of Arithmetic, we can factor $z$ as a product of one or more primes: :$z = p_1^{b_1} p_2^{b_2} \dots p_k^{b_k}$ Then for every prime number $q$: :$\norm z_q = \begin{cases} p_i^{-b_i} & : \exists i \in \closedint 1 k :q = p_i \\ 1 & : \forall i \in \closedint 1 k : q \ne p_i \\ \end {cases}$ By definition of absolute value on $\Q$: :$\size z = p_1^{b_1} p_2^{b_2} \dots p_k^{b_k} $ For $n \ge \max \set{p_1, p_2 \dots p_k}$: {{begin-eqn}} {{eqn | l = \size z \times \prod_{p \mathop \in \Bbb P \mathop : p \mathop \le n } \norm z_p       | r = \paren {p_1^{b_1} p_2^{b_2} \dots p_k^{b_k} } \times \paren {p_1^{-b_1} p_2^{-b_2} \dots p_k^{-b_k} }  }} {{eqn | r = 1 }} {{end-eqn}} Hence: {{begin-eqn}} {{eqn | l = \size z \times \prod_{p \mathop\in \Bbb P} \norm z_p       | r = \lim_{n \mathop \to \infty} \paren{\size z \times \prod_{p \mathop \in \Bbb P \mathop : p \mathop \le n} \norm z_p}       | c = {{Defof|Infinite Product}} }} {{eqn | r = 1       | c = Eventually Constant Sequence Converges to Constant }} {{end-eqn}} {{qed|lemma}} === Case 2 : $z \in \Z_{<0}$ === Let $z \in \Z_{<0}$. Hence : $-z \in \Z_{>0}$. We have: {{begin-eqn}} {{eqn | l = \size z \times \prod_{p \mathop\in \Bbb P} \norm z_p       | r = \size {-z} \times \prod_{p \mathop \in \Bbb P} \norm {-z}_p       | c = Norm of Negative }} {{eqn | r = 1       | c = Case 1 }} {{end-eqn}} {{qed|lemma}} In either case: :$\size z \times \displaystyle \prod_{p \mathop \in \Bbb P} \norm z_p = 1$ {{qed}} Category:P-adic Number Theory dwzm2ub4btf424j902li9t47zqm25fb
By definition, the $p$-adic numbers are the unique (up to isometric isomorphism) non-Archimedean valued field that completes $\struct {\Q, \norm {\,\cdot\,}_p}$ and $\norm {\,\cdot\,}_p$ is a non-Archimedean norm. The definition of an open ball in $p$-adic numbers is identical to the definition of an open ball of a normed division ring with respect to the norm $\norm {\,\cdot\,}_p$. {{qed}} Category:Definitions/Open Balls  Category:Definitions/P-adic Number Theory pet2kommg23356y64mnh4ttpfocrvep
By definition, the $p$-adic numbers are the unique (up to isometric isomorphism) non-Archimedean valued field that completes $\struct {\Q, \norm {\,\cdot\,}_p}$ and $\norm {\,\cdot\,}_p$ is a non-Archimedean norm. The definition of a closed ball in $p$-adic numbers is identical to the definition of a closed ball of a normed division ring with respect to the norm $\norm {\,\cdot\,}_p$. {{qed}} Category:Definitions/P-adic Number Theory 4vy5e5qipmoxnwxru1tx0wc988p7ujm
By definition, the $p$-adic numbers are the unique (up to isometric isomorphism) non-Archimedean valued field that completes $\struct {\Q, \norm {\,\cdot\,}_p}$ and $\norm {\,\cdot\,}_p$ is a non-Archimedean norm. The definition of a sphere in $p$-adic numbers is identical to the definition of a sphere in a normed division ring with respect to the norm $\norm {\,\cdot\,}_p$. {{qed}} Category:Definitions/P-adic Number Theory p2xvi2tweh0f7d100wnsm1lgz788tbn
The result follows directly from: :P-adic Closed Ball is Instance of Closed Ball of a Norm :P-adic Open Ball is Instance of Open Ball of a Norm :P-adic Sphere is Instance of Sphere of a Norm :Sphere is Set Difference of Closed and Open Ball in Normed Division Ring {{qed}} Category:P-adic Number Theory 4ssdrfo0qp1k4n27x56x7rjf0q63hay
By definition of a closed set in $M$: :$F$ is closed set in $M$ {{iff}} $A \setminus F$ is open in $M$ By definition of the topology $\tau$ induced by the metric $d$: :$A \setminus F$ is open in $M$ {{iff}} $A \setminus F$ is open in $\struct {A, \tau}$ By definition of a closed set in $\struct{A, \tau}$: :$A \setminus F$ is open in $\struct {A, \tau}$ {{iff}} $F$ is closed set in $\struct {A, \tau}$ The result follows. {{qed}} Category:Closed Sets Category:Metric Spaces l4ycvi1ahs62ph5bgwvds2evtt68snc
Let: {{begin-eqn}} {{eqn | l = u       | r = \cosh x }} {{eqn | ll= \leadsto       | l = u'       | r = \sinh x       | c = Derivative of Hyperbolic Cosine }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \int \csch x \rd x       | r = \int \frac 1 {\sinh x} \rd x       | c = {{Defof|Hyperbolic Cosecant}} }} {{eqn | r = \int \frac {\sinh x} {\sinh^2 x} \rd x       | c = multiplying top and bottom by $\sinh x$ }} {{eqn | r = \int \frac {\sinh x} {\cosh^2 x - 1} \rd x       | c = Difference of Squares of Hyperbolic Cosine and Sine }} {{eqn | r = \int \frac {\rd u} {u^2 - 1}       | c = Integration by Substitution }} {{eqn | r = -\coth^{-1} u + C       | c = Primitive of Reciprocal of $x^2 - a^2$: $\coth^{-1}$ form }} {{eqn | r = -\map {\coth^{-1} } {\cosh x} + C       | c = Definition of $u$ }} {{end-eqn}} {{qed}}
Let: {{begin-eqn}} {{eqn | l = u       | r = \cosh x       | c =  }} {{eqn | ll= \leadsto       | l = \dfrac {\d u} {\d x}       | r = \sinh x       | c = Derivative of Hyperbolic Cosine }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \int \csch x \rd x       | r = \int \dfrac {\d x} {\sinh x}       | c = {{Defof|Hyperbolic Cosecant|index = 2}} }} {{eqn | r = \int \dfrac 1 {\sinh x} \dfrac {\d u} {\sinh x}       | c = Integration by Substitution }} {{eqn | r = \int \dfrac {\d u} {\sinh^2 x}       | c = simplifying }} {{eqn | r = \int \dfrac {\d u} {\cosh^2 x - 1}       | c = Difference of Squares of Hyperbolic Cosine and Sine }} {{eqn | r = \int \dfrac {\d u} {u^2 - 1}       | c = Definition of $u$ }} {{eqn | r = -\coth^{-1} u + C       | c = Primitive of Reciprocal of $x^2 - a^2$: $\coth^{-1}$ form }} {{eqn | r = -\map {\coth^{-1} } {\cosh x} + C       | c = Definition of $u$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \int \frac 1 {\sqrt {a^2 - x^2} } \rd x       | r = \int \frac {\rd x} {\sqrt {a^2 \paren {1 - \frac {x^2} {a^2} } } }        | c = factor $a^2$ out of the radicand }} {{eqn | r = \int \frac {\rd x} {\sqrt{a^2} \sqrt {1 - \paren {\frac x a}^2} }       | c =  }} {{eqn | r = \frac 1 a \int \frac {\rd x} {\sqrt {1 - \paren {\frac x a}^2} } }} {{end-eqn}} Substitute: :$\cos \theta = \dfrac x a \iff x = a \cos \theta$ for $\theta \in \openint 0 \pi$. From Real Cosine Function is Bounded and Shape of Cosine Function, this substitution is valid for all $x / a \in \openint {-1} 1$. By hypothesis: {{begin-eqn}} {{eqn | l = a^2       | o = >        | r = x^2       | c =  }} {{eqn | ll= \leadstoandfrom       | l = 1       | o = >       | r = \frac {x^2} {a^2}       | c = dividing both terms by $a^2$ }} {{eqn | ll= \leadstoandfrom       | l = 1       | o = >       | r = \paren {\frac x a}^2       | c = Powers of Group Elements }} {{eqn | ll= \leadstoandfrom       | l = 1       | o = >       | r = \size {\frac x a}       | c = taking the square root of both terms }} {{eqn | ll= \leadstoandfrom       | l = -1       | o = <       | r = \paren {\frac x a} < 1       | c = Negative of Absolute Value }} {{end-eqn}} so this substitution will not change the domain of the integrand. Then: {{begin-eqn}} {{eqn | l = x        | r = a \cos \theta       | c = from above }} {{eqn | ll= \leadsto       | l = 1       | r = -a \sin \theta \frac {\rd \theta} {\rd x}       | c = differentiating {{WRT|Differentiation}} $x$, Derivative of Cosine Function, Chain Rule for Derivatives }} {{eqn | l = \frac 1 a \int \frac 1 {\sqrt {1 - \paren {\frac x a}^2 } } \rd x       | r = \frac 1 a \int \frac {-a \sin \theta} {\sqrt {1 - \cos^2 \theta} } \frac {\rd \theta} {\rd x} \rd x       | c = from above }} {{eqn | r = -\frac a a \int \frac {\sin \theta} {\sqrt {1 - \cos^2 \theta} } \rd \theta       | c = Integration by Substitution }} {{eqn | r = -\int \frac {\sin \theta} {\sqrt {\sin^2 \theta} } \rd \theta       | c = Sum of Squares of Sine and Cosine }} {{eqn | r = -\int \frac {\sin \theta} {\size {\sin \theta} } \rd \theta       | c =  }} {{end-eqn}} We have defined $\theta$ to be in the open interval $\openint 0 \pi$. From Sine and Cosine are Periodic on Reals, $\sin \theta > 0$ for the entire interval. Therefore the absolute value is unnecessary, and the integral simplifies to: {{begin-eqn}} {{eqn | l = -\int \rd \theta       | r = -\theta + C }} {{end-eqn}} As $\theta$ was stipulated to be in the open interval $\openint 0 \pi$: :$\cos \theta = \dfrac x a \iff \theta = \arccos \dfrac x a$ The answer in terms of $x$, then, is: :$\displaystyle \int \frac 1 {\sqrt {a^2 - x^2} } \rd x = -\arccos \frac x a + C$ {{qed}}
First we note that if $x > 1$ then $x + \sqrt {x^2 - a^2} > 0$. Hence $\map \ln {x + \sqrt {x^2 - a^2} }$ is defined. Then we have: {{begin-eqn}} {{eqn | l = -\map \ln {x + \sqrt {x^2 - a^2} }       | r = \map \ln {\dfrac 1 {x + \sqrt {x^2 - a^2} } }       | c = Logarithm of Reciprocal }} {{eqn | r = \map \ln {\dfrac {x - \sqrt {x^2 - a^2} } {\paren {x + \sqrt {x^2 - a^2} } \paren {x - \sqrt {x^2 - a^2} } } }       | c = multiplying top and bottom by $x - \sqrt {x^2 - a^2}$ }} {{eqn | r = \map \ln {\dfrac {x - \sqrt {x^2 - a^2} } {x^2 - \paren {x^2 - a^2} } }       | c = Difference of Two Squares }} {{eqn | r = \map \ln {\dfrac {x - \sqrt {x^2 - a^2} } {a^2} }       | c = simplifying }} {{eqn | r = \map \ln {x - \sqrt {x^2 - a^2} } - \map \ln {a^2}       | c = Difference of Logarithms }} {{end-eqn}} {{qed}}
{{Proofread}} Assume $y \in \R$, $ -\dfrac \pi 2 \le y \le \dfrac \pi 2 $. {{begin-eqn}} {{eqn | l = y       | r = \arccot x }} {{eqn | ll= \leadstoandfrom       | l = x       | r = \cot y }} {{eqn | ll= \leadstoandfrom       | l = x       | r = i \frac {1 + e^{2 i y} } {1 - e^{2 i y} }       | c = Cotangent Exponential Formulation }} {{eqn | ll= \leadstoandfrom       | l = i x       | r = \frac {e^{2 i y} + 1} {e^{2 i y} - 1}       | c = $ i^2 = -1 $ }} {{eqn | ll= \leadstoandfrom       | l = i x \paren {e^{2 i y} - 1}       | r = e^{2 i y} + 1 }} {{eqn | ll= \leadstoandfrom       | l = i x e^{2 i y} - i x       | r = e^{2 i y} + 1 }} {{eqn | ll= \leadstoandfrom       | l = e^{2 i y} - i x e^{2 i y}       | r = 1 + i x }} {{eqn | ll= \leadstoandfrom       | l = e^{2 i y}       | r = \frac {1 + i x} {1 - i x} }} {{eqn | ll= \leadstoandfrom       | l = y       | r = \frac 1 2 i \map \ln {\frac {1 + i x} {1 - i x} } }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn	| l = \int \frac {\d x} {x^2 + a^2} 	| r = \frac 1 a \int \frac {\d t} {t^2 + 1} 	| c = Substitution of $x \to a t$}} {{eqn	| r = \frac 1 a \int \frac {\d t} {\paren {1 + i t} \paren {1 - i t} } 	| c = Factoring }} {{eqn	| r = \frac 1 {2 a} \paren {\int \frac {\d t} {1 + i t} + \int \frac {\d t} {1 - i t} } 	| c = {{Defof|Partial Fractions Expansion}} }} {{eqn	| r = \frac 1 {2 a} \paren {i \map \ln {1 - i t} - i \map \ln {1 + i t} } + C 	| c = Primitive of Reciprocal }} {{eqn	| r = \frac i {2 a} \map \ln {\frac {1 - i t} {1 + i t} } + C 	| c = Sum of Logarithms }} {{eqn	| r = -\frac i {2 a} \map \ln {\frac {1 + i t} {1 - i t} } + C 	| c = Logarithm of Reciprocal }} {{eqn	| r = -\frac 1 a \arccot \frac x a + C 	| c = Arccotangent Logarithmic Formulation and substituting back $t \to \dfrac x a$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \int \frac {\d x} {x \sqrt {x^2 + a^2} }       | r = -\frac 1 a \map \ln {\frac {a + \sqrt {x^2 + a^2} } x} + C       | c = Primitive of Reciprocal of $x \sqrt {x^2 + a^2}$: Logarithm form }} {{eqn | r = \frac 1 a \map \ln {\frac x {a + \sqrt {a^2 + x^2} } } + C       | c = Logarithm of Reciprocal }} {{end-eqn}} {{qed}}
Let $y \in B_\epsilon \left({x}\right)$. From Open Ball of Point Inside Open Ball, there exists $\delta \in \R_{>0}$ such that $B_\delta \left({y}\right) \subseteq B_\epsilon \left({x}\right)$ The result follows from the definition of open set. {{qed}}
{{begin-eqn}} {{eqn | l = \map {\dfrac \d {\d x} } {\cosh x}       | r = \map {\dfrac \d {\d x} } {\dfrac {e^x + e ^{-x} } 2}       | c = {{Defof|Hyperbolic Cosine}} }} {{eqn | r = \dfrac 1 2 \map {\dfrac \d {\d x} } {e^x + e^{-x} }       | c = Derivative of Constant Multiple }} {{eqn | r = \dfrac 1 2 \paren {e^x + \paren {-e^{-x} } }       | c = Derivative of Exponential Function, Chain Rule for Derivatives, Linear Combination of Derivatives }} {{eqn | r = \dfrac {e^x - e^{-x} } 2 }} {{eqn | r = \sinh x       | c = {{Defof|Hyperbolic Sine}} }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \map {\dfrac \d {\d x} } {\tanh x}       | r = \map {\dfrac \d {\d x} } {\dfrac {\sinh x} {\cosh x} }       | c = {{Defof|Hyperbolic Tangent|index = 2}} }} {{eqn | r = \dfrac {\paren {\dfrac \d {\d x}  \sinh x} \cosh x - \sinh x \paren {\dfrac \d {\d x}  \cosh x} } {\cosh^2 x}       | c = Quotient Rule for Derivatives }} {{eqn | r = \dfrac {\cosh^2 x - \sinh x \paren {\dfrac \d {\d x} \cosh x} } {\cosh^2 x}       | c = Derivative of Hyperbolic Sine }} {{eqn | r = \dfrac {\cosh^2 x - \sinh^2 x} {\cosh^2 x}       | c = Derivative of Hyperbolic Cosine }} {{eqn | r = \dfrac 1 {\cosh^2 x}       | c = Difference of Squares of Hyperbolic Cosine and Sine }} {{eqn | r = \sech^2 x       | c = {{Defof|Hyperbolic Secant|index = 2}} }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \map {\dfrac \d {\d x} } {\coth x}       | r = \map {\dfrac \d {\d x} } {\frac {\cosh x} {\sinh x} }       | c = {{Defof|Hyperbolic Cotangent|index = 2}} }} {{eqn | r = \frac {\sinh x \dfrac \d {\d x} \cosh x - \cosh x \dfrac \d {\d x} \sinh x} {\sinh^2 x}       | c = Quotient Rule for Derivatives }} {{eqn | r = \frac {\sinh x \sinh x - \cosh x \dfrac \d {\d x} \cosh x} {\sinh^2 x}       | c = Derivative of Hyperbolic Cosine }} {{eqn | r = \frac {\sinh x \sinh x - \cosh x \cosh x} {\sinh^2 x}       | c = Derivative of Hyperbolic Sine }} {{eqn | r = \frac {-1} {\sinh^2 x}       | c = Difference of Squares of Hyperbolic Cosine and Sine }} {{eqn | r = -\csch^2 x       | c = {{Defof|Hyperbolic Cosecant|index = 2}} }} {{end-eqn}} {{qed}}
It is noted that at $x = 0$, $\csch x$ is undefined. Hence the restriction of the domain. {{begin-eqn}} {{eqn | l = \map {\dfrac \d {\d x} } {\csch x}       | r = \map {\dfrac \d {\d x} } {\frac 1 {\sinh x} }       | c = {{Defof|Hyperbolic Cosecant}} }} {{eqn | r = \map {\dfrac \d {\d x} } {\paren {\sinh z}^{-1} }       | c = Exponent Laws }} {{eqn | r = -\paren {\sinh x}^{-2} \cosh x       | c = Derivative of Hyperbolic Cosine, Power Rule for Derivatives, Chain Rule for Derivatives }} {{eqn | r = \frac {-1} {\sinh x} \ \frac {\cosh x} {\sinh x}       | c = Exponent Combination Laws }} {{eqn | r = -\csch x \coth x       | c = {{Defof|Hyperbolic Cosecant}} and {{Defof|Hyperbolic Cotangent}} }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \map {\frac \d {\d x} } {\sinh^{-1} u}       | r = \map {\frac \d {\d u} } {\sinh^{-1} u} \frac {\d u} {\d x}       | c = Chain Rule for Derivatives }} {{eqn | r = \dfrac 1 {\sqrt {1 + u^2} } \frac {\d u} {\d x}       | c = Derivative of Inverse Hyperbolic Sine }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \map {\frac \d {\d x} } {\cosh^{-1} u}       | r = \map {\frac \d {\d u} } {\cosh^{-1} u} \frac {\d u} {\d x}       | c = Chain Rule for Derivatives }} {{eqn | r = \dfrac 1 {\sqrt {u^2 - 1} } \frac {\d u} {\d x}       | c = Derivative of Inverse Hyperbolic Cosine }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \map {\frac \d {\d x} } {\tanh^{-1} u}       | r = \map {\frac \d {\d u} } {\tanh^{-1} u} \frac {\d u} {\d x}       | c = Chain Rule for Derivatives }} {{eqn | r = \dfrac 1 {1 - u^2} \frac {\d u} {\d x}       | c = Derivative of Inverse Hyperbolic Tangent }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \map {\frac \d {\d x} } {\coth^{-1} u}       | r = \map {\frac \d {\d u} } {\coth^{-1} u} \frac {\d u} {\d x}       | c = Chain Rule for Derivatives }} {{eqn | r = \dfrac {-1} {u^2 - 1} \frac {\d u} {\d x}       | c = Derivative of Inverse Hyperbolic Cotangent }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \map {\frac \d {\d x} } {\sech^{-1} u}       | r = \map {\frac \d {\d u} } {\sech^{-1} u} \frac {\d u} {\d x}       | c = Chain Rule for Derivatives }} {{eqn | r = \dfrac {-1} {u \sqrt {1 - u^2} } \frac {\d u} {\d x}       | c = Derivative of Inverse Hyperbolic Secant }} {{end-eqn}} {{qed}}
600pxright {{begin-eqn}} {{eqn | l = \map {\frac \d {\d x} } {\csch^{-1} u}       | r = \map {\frac \d {\d u} } {\csch^{-1} u} \frac {\d u} {\d x}       | c = Chain Rule for Derivatives }} {{eqn | r = \dfrac {-1} {\size u \sqrt {1 + u^2} } \frac {\d u} {\d x}       | c = Derivative of Inverse Hyperbolic Cosecant }} {{end-eqn}} {{qed}}
Let $I$ be an indexing set (either finite or infinite). Let $\displaystyle \bigcap_{i \mathop \in I} V_i$ be the intersection of a indexed family of closed sets of $T$ indexed by $I$. Then from De Morgan's laws: Difference with Intersection: :$\displaystyle S \setminus \bigcap_{i \mathop \in I} V_i = \bigcup_{i \mathop \in I} \paren {S \setminus V_i}$ By definition of closed set, each of $S \setminus V_i$ are by definition open in $T$. We have that $\displaystyle \bigcup_{i \mathop \in I} \paren {S \setminus V_i}$ is the union of a indexed family of open sets of $T$ indexed by $I$. Therefore, by definition of a topology, $\displaystyle \bigcup_{i \mathop \in I} \paren {S \setminus V_i} = S \setminus \bigcap_{i \mathop \in I} V_i$ is likewise open in $T$. Then by definition of closed set, $\displaystyle \bigcap_{i \mathop \in I} V_i$ is closed in $T$. {{qed}}
Let $\displaystyle \bigcup_{i \mathop = 1}^n V_i$ be the union of a finite number of closed sets of $T$. Then from De Morgan's laws: :$\displaystyle S \setminus \bigcup_{i \mathop = 1}^n V_i = \bigcap_{i \mathop = 1}^n \left({S \setminus V_i}\right)$ By definition of closed set, each of the $S \setminus V_i$ is by definition open in $T$. We have that $\displaystyle \bigcap_{i \mathop = 1}^n \left({S \setminus V_i}\right)$ is the intersection of a finite number of open sets of $T$. Therefore, by definition of a topology, $\displaystyle \bigcap_{i \mathop = 1}^n \left({S \setminus V_i}\right) = S \setminus \bigcup_{i \mathop = 1}^n V_i$ is likewise open in $T$. Then by definition of closed set, $\displaystyle \bigcup_{i \mathop = 1}^n V_i$ is closed in $T$. {{qed}}
We show that the complement $A \setminus B_\epsilon^- \left({x}\right)$ is open in $M$. Let $a \in A \setminus \map {B_\epsilon^-} x$. Then by definition of closed ball: :$\map d {x, a} > \epsilon$ Put: :$\delta := \map d {x, a} - \epsilon > 0$ Then: :$\map d {x, a} - \delta = \epsilon$ Let $b \in \map {B_\delta} a$. Then: {{begin-eqn}} {{eqn | l = \map d {x, b}       | o = \ge        | r = \map d {x, a} - \map d {a, b}       | c = Reverse Triangle Inequality }} {{eqn | o = >       | r = \map d {x, a} - \delta }} {{eqn | r = \epsilon }} {{end-eqn}} and so: :$b \notin \map {B_\epsilon^-} x$ Then: :$\map {B_\delta} a \subseteq A \setminus \map {B_\epsilon^-} x$ so $A \setminus \map {B_\epsilon^-} x$ is open in $M$. Hence, by definition of closed set: :$\map {B_\epsilon^-} x$ is closed in $M$. {{qed}} Category:Metric Spaces Category:Closed Balls i0t1kn6zyg0qs0ijte71kxgpqtx78ez
{{begin-eqn}} {{eqn | l = \map {\frac \d {\d x} } {\sin u}       | r = \map {\frac \d {\d u} } {\sin u} \frac {\d u} {\d x}       | c = Chain Rule for Derivatives }} {{eqn | r = \cos u \frac {\d u} {\d x}       | c = Derivative of Sine Function }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \map {\frac \d {\d x} } {\cos u}       | r = \map {\frac \d {\d u} } {\cos u} \frac {\d u} {\d x}       | c = Chain Rule for Derivatives }} {{eqn | r = -\sin u \frac {\d u} {\d x}       | c = Derivative of Cosine Function }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \map {\frac \d {\d x} } {\tan u}       | r = \map {\frac \d {\d u} } {\tan u} \frac {\d u} {\d x}       | c = Chain Rule for Derivatives }} {{eqn | r = \sec^2 u \frac {\d u} {\d x}       | c = Derivative of Tangent Function }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \map {\frac \d {\d x} } {\cot u}       | r = \map {\frac \d {\d u} } {\cot u} \frac {\d u} {\d x}       | c = Chain Rule for Derivatives }} {{eqn | r = -\csc^2 u \frac {\d u} {\d x}       | c = Derivative of Cotangent Function }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \map {\frac \d {\d x} } {\sec u}       | r = \map {\frac \d {\d u} } {\sec u} \frac {\d u} {\d x}       | c = Chain Rule for Derivatives }} {{eqn | r = \sec u \tan u \frac {\d u} {\d x}       | c = Derivative of Secant Function }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \map {\frac \d {\d x} } {\csc u}       | r = \map {\frac \d {\d u} } {\csc u} \frac {\d u} {\d x}       | c = Chain Rule for Derivatives }} {{eqn | r = -\csc u \cot u \frac {\d u} {\d x}       | c = Derivative of Cosecant Function }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \map {\frac \d {\d x} } {\log_a u}       | r = \map {\frac \d {\d u} } {\log_a u} \frac {\d u} {\d x}       | c = Chain Rule for Derivatives }} {{eqn | r = \dfrac {\log_a e} u \frac {\d u} {\d x}       | c = Derivative of General Logarithm Function }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \map {\frac \d {\d x} } {\ln u}       | r = \map {\frac \d {\d u} } {\ln u} \frac {\d u} {\d x}       | c = Chain Rule for Derivatives }} {{eqn | r = \dfrac 1 u \frac {\d u} {\d x}       | c = Derivative of Natural Logarithm Function }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \map {\frac \d {\d x} } {a^u}       | r = \map {\frac \d {\d u} } {a^u} \frac {\d u} {\d x}       | c = Chain Rule for Derivatives }} {{eqn | r = a^u \ln a \frac {\d u} {\d x}       | c = Derivative of Power of Constant }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \map {\frac \d {\d x} } {e^u}       | r = \map {\frac \d {\d u} } {e^u} \frac {\d u} {\d x}       | c = Chain Rule for Derivatives }} {{eqn | r = e^u \frac {\d u} {\d x}       | c = Derivative of Exponential Function }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \map {\frac \d {\d x} } {\arcsin u}       | r = \map {\frac \d {\d u} } {\arcsin u} \frac {\d u} {\d x}       | c = Chain Rule for Derivatives }} {{eqn | r = \dfrac 1 {\sqrt {1 - u^2} } \frac {\d u} {\d x}       | c = Derivative of Arcsine Function }} {{end-eqn}} {{qed}}
{{:Graph of Arccosine Function}} {{begin-eqn}} {{eqn | l = \map {\frac \d {\d x} } {\arccos u}       | r = \map {\frac \d {\d u} } {\arccos u} \frac {\d u} {\d x}       | c = Chain Rule for Derivatives }} {{eqn | r = -\dfrac 1 {\sqrt {1 - u^2} } \frac {\d u} {\d x}       | c = Derivative of Arccosine Function }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \map {\frac \d {\d x} } {\arctan u}       | r = \map {\frac \d {\d u} } {\arctan u} \frac {\d u} {\d x}       | c = Chain Rule for Derivatives }} {{eqn | r = \dfrac 1 {1 + u^2} \frac {\d u} {\d x}       | c = Derivative of Arctangent Function }} {{end-eqn}} {{qed}}