nyuuzyou/wikis huggingface dataset:
Combination Theorem for Limits of Functions/Real/Sum Rule

== Theorem ==
Let $\R$ denote the real numbers.
Let $f$ and $g$ be real functions defined on an open subset $S \subseteq \R$, except possibly at the point $c \in S$.
Let $f$ and $g$ tend to the following limits:
:$\ds \lim_{x \mathop \to c} \map f x = l$
:$\ds \lim_{x \mathop \to c} \map g x = m$
Then:
:$\ds \lim_{x \mathop \to c} \paren {\map f x + \map g x} = l + m$
== Proof ==
Let $\sequence {x_n}$ be any sequence of elements of $S$ such that:
:$\forall n \in \N_{>0}: x_n \ne c$
:$\ds \lim_{n \mathop \to \infty} \ x_n = c$
By Limit of Real Function by Convergent Sequences:
:$\ds \lim_{n \mathop \to \infty} \map f {x_n} = l$
:$\ds \lim_{n \mathop \to \infty} \map g {x_n} = m$
By the Sum Rule for Real Sequences:
:$\ds \lim_{n \mathop \to \infty} \paren {\map f {x_n} + \map g {x_n} } = l + m$
Applying Limit of Real Function by Convergent Sequences again, we get:
:$\ds \lim_{x \mathop \to c} \paren {\map f x + \map g x} = l + m$
== Sources ==
* : Chapter $\text I$: Functions and Limits: $\S 4$: Limits of Functions: Theorem $1 \ \text{(i)}$
* : $1$: Review of some real analysis: $\S 1.4$: Continuity
Category:Combination Theorems for Limits of Functions
Combination Theorem for Limits of Functions/Sum Rule

== Theorem ==
=== Real Functions ===
Let $\R$ denote the real numbers.
Let $f$ and $g$ be real functions defined on an open subset $S \subseteq \R$, except possibly at the point $c \in S$.
Let $f$ and $g$ tend to the following limits:
:$\ds \lim_{x \mathop \to c} \map f x = l$
:$\ds \lim_{x \mathop \to c} \map g x = m$
Then:
=== Complex Functions ===
Let $\C$ denote the complex numbers.
Let $f$ and $g$ be complex functions defined on an open subset $S \subseteq \C$, except possibly at the point $c \in S$.
Let $f$ and $g$ tend to the following limits:
:$\ds \lim_{z \mathop \to c} \map f z = l$
:$\ds \lim_{z \mathop \to c} \map g z = m$
Then:
Category:Combination Theorems for Limits of Functions
Combination Theorem for Limits of Mappings/Metric Space

== Theorem ==
Let $M = \struct {A, d}$ be a metric space.
Let $\R$ denote the real numbers.
Let $f: A \to \R$ and $g: A \to \R$ be real-valued functions defined on $A$, except possibly at the point $a \in A$.
Let $f$ and $g$ tend to the following limits:
:$\ds \lim_{x \mathop \to a} \map f x = l$
:$\ds \lim_{x \mathop \to a} \map g x = m$
Let $\lambda, \mu \in \R$ be arbitrary real numbers.
Then the following results hold:
=== Sum Rule ===
=== Multiple Rule ===
=== Product Rule ===
=== Quotient Rule ===
Category:Metric Spaces
Category:Combination Theorem for Limits of Mappings
Category:Combination Theorem for Limits of Mappings in Metric Spaces
Combination Theorem for Limits of Mappings/Metric Space/Multiple Rule

== Theorem ==
Let $M = \struct {A, d}$ be a metric space.
Let $\R$ denote the real numbers.
Let $f: A \to \R$ be a real-valued function defined on $A$, except possibly at the point $a \in A$.
Let $f$ tend to the following limit:
:$\ds \lim_{x \mathop \to a} \map f x = l$
Let $\lambda \in \R$ be an arbitrary real number.
Then:
:$\ds \lim_{x \mathop \to a} \lambda \map f x = \lambda l$
== Proof ==
Let $\sequence {x_n}$ be any sequence of elements of $S$ such that:
:$\forall n \in \N_{>0}: x_n \ne a$
:$\ds \lim_{n \mathop \to \infty} \ x_n = a$
By Limit of Function by Convergent Sequences:
:$\ds \lim_{n \mathop \to \infty} \map f {x_n} = l$
By the Multiple Rule for Real Sequences:
:$\ds \lim_{n \mathop \to \infty} \lambda \map f {x_n} = \lambda l$
Applying Limit of Function by Convergent Sequences again, we get:
:$\ds \lim_{x \mathop \to a} \lambda \map f x = \lambda l$
</onlyinclude>
Category:Combination Theorem for Limits of Mappings in Metric Spaces
Combination Theorem for Limits of Mappings/Metric Space/Product Rule

== Theorem ==
Let $M = \struct {A, d}$ be a metric space.
Let $\R$ denote the real numbers.
Let $f: A \to \R$ and $g: A \to \R$ be real-valued functions defined on $A$, except possibly at the point $a \in A$.
Let $f$ and $g$ tend to the following limits:
:$\ds \lim_{x \mathop \to a} \map f x = l$
:$\ds \lim_{x \mathop \to a} \map g x = m$
Then:
:$\ds \lim_{x \mathop \to a} \paren {\map f x \map g x} = l m$
== Proof ==
Let $\sequence {x_n}$ be any sequence of elements of $S$ such that:
:$\forall n \in \N_{>0}: x_n \ne a$
:$\ds \lim_{n \mathop \to \infty} \ x_n = a$
By Limit of Function by Convergent Sequences:
:$\ds \lim_{n \mathop \to \infty} \map f {x_n} = l$
:$\ds \lim_{n \mathop \to \infty} \map g {x_n} = m$
By the Product Rule for Real Sequences:
:$\ds \lim_{n \mathop \to \infty} \paren {\map f {x_n} \map g {x_n} } = l m$
Applying Limit of Function by Convergent Sequences again, we get:
:$\ds \lim_{x \mathop \to a} \paren {\map f x \map g x} = l m$
</onlyinclude>
Category:Combination Theorem for Limits of Mappings in Metric Spaces
Combination Theorem for Limits of Mappings/Metric Space/Quotient Rule

== Theorem ==
Let $M = \struct {A, d}$ be a metric space.
Let $\R$ denote the real numbers.
Let $f: A \to \R$ and $g: A \to \R$ be real-valued functions defined on $A$, except possibly at the point $a \in A$.
Let $f$ and $g$ tend to the following limits:
:$\ds \lim_{x \mathop \to a} \map f x = l$
:$\ds \lim_{x \mathop \to a} \map g x = m$
Then:
:$\ds \lim_{x \mathop \to a} \frac {\map f x} {\map g x} = \frac l m$
provided that $m \ne 0$.
== Proof ==
Let $\sequence {x_n}$ be any sequence of elements of $S$ such that:
:$\forall n \in \N_{>0}: x_n \ne a$
:$\ds \lim_{n \mathop \to \infty} \ x_n = a$
By Limit of Function by Convergent Sequences:
:$\ds \lim_{n \mathop \to \infty} \map f {x_n} = l$
:$\ds \lim_{n \mathop \to \infty} \map g {x_n} = m$
By the Quotient Rule for Real Sequences:
:$\ds \lim_{n \mathop \to \infty} \frac {\map f {x_n} } {\map g {x_n} } = \frac l m$
provided that $m \ne 0$.
Applying Limit of Function by Convergent Sequences again, we get:
:$\ds \lim_{x \mathop \to a} \frac {\map f x} {\map g x} = \frac l m$
provided that $m \ne 0$.
</onlyinclude>
Category:Combination Theorem for Limits of Mappings in Metric Spaces
Combination Theorem for Limits of Mappings/Metric Space/Sum Rule

== Theorem ==
Let $M = \struct {A, d}$ be a metric space.
Let $\R$ denote the real numbers.
Let $f: A \to \R$ and $g: A \to \R$ be real-valued functions defined on $A$, except possibly at the point $a \in A$.
Let $f$ and $g$ tend to the following limits:
:$\ds \lim_{x \mathop \to a} \map f x = l$
:$\ds \lim_{x \mathop \to a} \map g x = m$
Then:
:$\ds \lim_{x \mathop \to a} \paren {\map f x + \map g x} = l + m$
== Proof ==
Let $\sequence {x_n}$ be any sequence of elements of $S$ such that:
:$\forall n \in \N_{>0}: x_n \ne a$
:$\ds \lim_{n \mathop \to \infty} \ x_n = a$
By Limit of Function by Convergent Sequences:
:$\ds \lim_{n \mathop \to \infty} \map f {x_n} = l$
:$\ds \lim_{n \mathop \to \infty} \map g {x_n} = m$
By the Sum Rule for Real Sequences:
:$\ds \lim_{n \mathop \to \infty} \paren {\map f {x_n} + \map g {x_n} } = l + m$
Applying Limit of Function by Convergent Sequences again, we get:
:$\ds \lim_{x \mathop \to a} \paren {\map f x + \map g x} = l + m$
</onlyinclude>
Category:Combination Theorem for Limits of Mappings in Metric Spaces
Combination Theorem for Sequences/Combined Sum Rule

== Theorem ==
=== Combined Sum Rule for Real Sequences ===
Let $\sequence {x_n}$ and $\sequence {y_n}$ be sequences in $\R$.
Let $\sequence {x_n}$ and $\sequence {y_n}$ be convergent to the following limits:
:$\ds \lim_{n \mathop \to \infty} x_n = l$
:$\ds \lim_{n \mathop \to \infty} y_n = m$
Let $\lambda, \mu \in \R$.
Then:
=== Combined Sum Rule for Complex Sequences ===
Let $\sequence {z_n}$ and $\sequence {w_n}$ be sequences in $\C$.
Let $\sequence {z_n}$ and $\sequence {w_n}$ be convergent to the following limits:
:$\ds \lim_{n \mathop \to \infty} z_n = c$
:$\ds \lim_{n \mathop \to \infty} w_n = d$
Let $\lambda, \mu \in \C$.
Then:
Category:Combination Theorems for Sequences
Combination Theorem for Sequences/Complex

== Theorem == 
Let $\sequence {z_n}$ and $\sequence {w_n}$ be sequences in $\C$.
Let $\sequence {z_n}$ and $\sequence {w_n}$ be convergent to the following limits:
:$\ds \lim_{n \mathop \to \infty} z_n = c$
:$\ds \lim_{n \mathop \to \infty} w_n = d$
Let $\lambda, \mu \in \C$.
Then the following results hold:
=== Sum Rule ===
=== Difference Rule ===
=== Multiple Rule ===
=== Combined Sum Rule ===
=== Product Rule ===
=== Quotient Rule ===
Category:Combination Theorems for Sequences
Category:Complex Analysis
Combination Theorem for Sequences/Complex/Combined Sum Rule

== Theorem == 
Let $\sequence {z_n}$ and $\sequence {w_n}$ be sequences in $\C$.
Let $\sequence {z_n}$ and $\sequence {w_n}$ be convergent to the following limits:
:$\ds \lim_{n \mathop \to \infty} z_n = c$
:$\ds \lim_{n \mathop \to \infty} w_n = d$
Let $\lambda, \mu \in \C$.
Then:
:$\ds \lim_{n \mathop \to \infty} \paren {\lambda z_n + \mu w_n} = \lambda c + \mu d$
== Proof ==
From the Multiple Rule for Complex Sequences, we have:
:$\ds \lim_{n \mathop \to \infty} \paren {\lambda z_n} = \lambda c$
:$\ds \lim_{n \mathop \to \infty} \paren {\mu w_n} = \mu d$
The result now follows directly from the Sum Rule for Complex Sequences:
:$\ds \lim_{n \mathop \to \infty} \paren {\lambda z_n + \mu w_n} = \lambda c + \mu d$
Category:Combination Theorems for Sequences
Combination Theorem for Sequences/Complex/Difference Rule

== Theorem == 
Let $\sequence {z_n}$ and $\sequence {w_n}$ be sequences in $\C$.
Let $\sequence {z_n}$ and $\sequence {w_n}$ be convergent to the following limits:
:$\ds \lim_{n \mathop \to \infty} z_n = c$
:$\ds \lim_{n \mathop \to \infty} w_n = d$
Then:
:$\ds \lim_{n \mathop \to \infty} \paren {z_n - w_n} = c - d$
== Proof ==
From Sum Rule for Complex Sequences:
:$\ds \lim_{n \mathop \to \infty} \paren {z_n + w_n} = c + d$
From Multiple Rule for Complex Sequences:
:$\ds \lim_{n \mathop \to \infty} \paren {-w_n} = -d$
Hence:
:$\ds \lim_{n \mathop \to \infty} \paren {z_n + \paren {-w_n} } = c + \paren {-d}$
The result follows.
== Also see ==
* Sum Rule for Complex Sequences
== Sources ==
* : $\S 4.2$. Sequences: Rules. $\text {(ii)}$
Category:Combination Theorems for Sequences
Combination Theorem for Sequences/Complex/Multiple Rule

== Theorem == 
Let $\sequence {z_n}$ be a sequence in $\C$.
Let $\sequence {z_n}$ be convergent to the following limit:
:$\ds \lim_{n \mathop \to \infty} z_n = c$
Let $\lambda \in \C$.
Then:
:$\ds \lim_{n \mathop \to \infty} \paren {\lambda z_n} = \lambda c$
== Proof ==
Let $\epsilon > 0$.
We need to find $N$ such that:
:$\forall n > N: \cmod {\lambda z_n - \lambda c} < \epsilon$
If $\lambda = 0$ the result is trivial.
So, assume $\lambda \ne 0$.
Then $\cmod \lambda > 0$ from the definition of the modulus of $\lambda$.
Hence $\dfrac \epsilon {\cmod \lambda} > 0$.
We have that $z_n \to c$ as $n \to \infty$.
Thus it follows that:
:$\exists N: \forall n > N: \cmod {z_n - c} < \dfrac \epsilon {\cmod \lambda}$
That is:
:$\forall n > N: \cmod \lambda \cmod {z_n - c} < \epsilon$
But we have:
Hence:
:$\ds \lim_{n \mathop \to \infty} \paren {\lambda x_n} = \lambda c$
Category:Combination Theorems for Sequences
Combination Theorem for Sequences/Complex/Product Rule

== Theorem == 
Let $\sequence {z_n}$ and $\sequence {w_n}$ be sequences in $\C$.
Let $\sequence {z_n}$ and $\sequence {w_n}$ be convergent to the following limits:
:$\ds \lim_{n \mathop \to \infty} z_n = c$
:$\ds \lim_{n \mathop \to \infty} w_n = d$
Then:
:$\ds \lim_{n \mathop \to \infty} \paren {z_n w_n} = c d$
== Proof 1 ==
== Proof 2 ==
== Sources ==
* : $\S 4.2$. Sequences: Rules. $\text {(iii)}$
Category:Combination Theorems for Sequences
Category:Product Rule for Complex Sequences
Combination Theorem for Sequences/Complex/Product Rule/Proof 1

== Theorem == 
== Proof ==
Because $\sequence {z_n}$ converges, it is bounded by Convergent Sequence is Bounded.
Suppose $\cmod {z_n} \le K$ for $n = 1, 2, 3, \ldots$.
Then:
But $z_n \to c$ as $n \to \infty$.
So $\cmod {z_n - c} \to 0$ as $n \to \infty$ from Convergent Sequence Minus Limit.
Similarly $\cmod {w_n - d} \to 0$ as $n \to \infty$.
From the Combined Sum Rule for Real Sequences:
:$\ds \lim_{n \mathop \to \infty} \paren {\lambda z_n + \mu w_n} = \lambda c + \mu d$, $\phi_n \to 0$ as $n \to \infty$
The result follows by the Squeeze Theorem for Complex Sequences.
Category:Product Rule for Complex Sequences
Combination Theorem for Sequences/Complex/Product Rule/Proof 2

== Theorem == 
== Proof ==
Let $z_n = x_n + i y_n$.
Let $w_n = u_n + i v_n$.
Let $c = a + i b$
Let $d = e + i f$.
By definition of convergent complex sequence:
Then:
== Sources ==
* : $\S 4.2$. Sequences: Rules
Category:Product Rule for Complex Sequences
Combination Theorem for Sequences/Complex/Quotient Rule

== Theorem == 
Let $\sequence {z_n}$ and $\sequence {w_n}$ be sequences in $\C$.
Let $\sequence {z_n}$ and $\sequence {w_n}$ be convergent to the following limits:
:$\ds \lim_{n \mathop \to \infty} z_n = c$
:$\ds \lim_{n \mathop \to \infty} w_n = d$
Then:
:$\ds \lim_{n \mathop \to \infty} \frac {z_n} {w_n} = \frac c d$
provided that $d \ne 0$.
== Proof ==
As $z_n \to c$ as $n \to \infty$, it follows from Modulus of Limit that $\size {w_n} \to \size d$ as $n \to \infty$.
As $d \ne 0$, it follows from the definition of the modulus of $d$ that $\size d > 0$.
From Sequence Converges to Within Half Limit, we have $\exists N: \forall n > N: \size {w_n} > \dfrac {\size d} 2$.
Now, for $n > N$, consider:
By the above, $d z_n - w_n c \to d c - d c = 0$ as $n \to \infty$.
The result follows by the Squeeze Theorem for Complex Sequences (which applies as well to real as to complex sequences).
== Also presented as ==
== Sources ==
* : $\S 4.2$. Sequences: Rules. $\text {(iv)}$
Category:Combination Theorems for Sequences
Combination Theorem for Sequences/Complex/Sum Rule

== Theorem == 
Let $\sequence {z_n}$ and $\sequence {w_n}$ be sequences in $\C$.
Let $\sequence {z_n}$ and $\sequence {w_n}$ be convergent to the following limits:
:$\ds \lim_{n \mathop \to \infty} z_n = c$
:$\ds \lim_{n \mathop \to \infty} w_n = d$
Then:
:$\ds \lim_{n \mathop \to \infty} \paren {z_n + w_n} = c + d$
== Proof 1 ==
== Proof 2 ==
== Also see ==
* Difference Rule for Complex Sequences
== Sources ==
* : $\S 4.2$. Sequences: Rules. $\text {(i)}$
Category:Combination Theorems for Sequences
Category:Sum Rule for Complex Sequences
Combination Theorem for Sequences/Complex/Sum Rule/Proof 1

== Theorem == 
Let $\sequence {z_n}$ and $\sequence {w_n}$ be sequences in $\C$.
Let $\sequence {z_n}$ and $\sequence {w_n}$ be convergent to the following limits:
:$\ds \lim_{n \mathop \to \infty} z_n = c$
:$\ds \lim_{n \mathop \to \infty} w_n = d$
Then:
== Proof ==
Let $\epsilon > 0$ be given.
Then $\dfrac \epsilon 2 > 0$.
We are given that:
:$\ds \lim_{n \mathop \to \infty} z_n = c$
:$\ds \lim_{n \mathop \to \infty} w_n = d$
By definition of the limit of a complex sequence, we can find $N_1$ such that:
:$\forall n > N_1: \cmod {z_n - c} < \dfrac \epsilon 2$
where $\cmod {z_n - c}$ denotes the complex modulus of $z_n - c$.
Similarly we can find $N_2$ such that:
:$\forall n > N_2: \cmod {w_n - d} < \dfrac \epsilon 2$
Let $N = \max \set {N_1, N_2}$.
Then if $n > N$, both the above inequalities will be true:
:$n > N_1$
:$n > N_2$
Thus $\forall n > N$:
Hence the result:
:$\ds \lim_{n \mathop \to \infty} \paren {z_n + w_n} = c + d$
Category:Sum Rule for Complex Sequences
Combination Theorem for Sequences/Complex/Sum Rule/Proof 2

== Theorem == 
Let $\sequence {z_n}$ and $\sequence {w_n}$ be sequences in $\C$.
Let $\sequence {z_n}$ and $\sequence {w_n}$ be convergent to the following limits:
:$\ds \lim_{n \mathop \to \infty} z_n = c$
:$\ds \lim_{n \mathop \to \infty} w_n = d$
Then:
== Proof ==
Let $\epsilon > 0$ be given.
Then $\dfrac \epsilon 2 > 0$.
We are given that:
:$\ds \lim_{n \mathop \to \infty} z_n = c$
:$\ds \lim_{n \mathop \to \infty} w_n = d$
Let:
:$z_n = x_n + i y_n$
:$w_n = r_n + i s_n$
:$c = a + i b$
:$d = l + i m$
where each of $x_n, y_n, r_n, s_n, a, b, l, m \in \R$ are real.
By definition:
$\sequence {z_n}$ converges to the limit $c = a + i b$ both:
:$\forall \epsilon \in \R_{>0}: \exists N \in \R: n > N \implies \size {x_n - a} < \epsilon \text { and } \size {y_n - b} < \epsilon$
$\sequence {w_n}$ converges to the limit $d = l + i m$ both:
:$\forall \epsilon \in \R_{>0}: \exists N \in \R: n > N \implies \size {r_n - l} < \epsilon \text { and } \size {s_n - m} < \epsilon$
where $\size x$ denotes the absolute value of $x \in \R$.
Then:
Category:Sum Rule for Complex Sequences
Combination Theorem for Sequences/Difference Rule

== Theorem ==
=== Real Sequences ===
Let $\sequence {x_n}$ and $\sequence {y_n}$ be sequences in $\R$.
Let $\sequence {x_n}$ and $\sequence {y_n}$ be convergent to the following limits:
:$\ds \lim_{n \mathop \to \infty} x_n = l$
:$\ds \lim_{n \mathop \to \infty} y_n = m$
Then:
=== Complex Sequences ===
Let $\sequence {z_n}$ and $\sequence {w_n}$ be sequences in $\C$.
Let $\sequence {z_n}$ and $\sequence {w_n}$ be convergent to the following limits:
:$\ds \lim_{n \mathop \to \infty} z_n = c$
:$\ds \lim_{n \mathop \to \infty} w_n = d$
Then:
== Also see ==
* Sum Rule for Sequences
* Multiple Rule for Sequences
* Product Rule for Sequences
* Quotient Rule for Sequences
== Sources ==
* 
* 
Category:Combination Theorems for Sequences
Combination Theorem for Sequences/Multiple Rule

== Theorem ==
=== Real Sequences ===
Let $\sequence {x_n}$ be a sequences in $\R$.
Let $\sequence {x_n}$ be convergent to the following limit:
:$\ds \lim_{n \mathop \to \infty} x_n = l$
Let $\lambda \in \R$.
Then:
=== Complex Sequences ===
Let $\sequence {z_n}$ be a sequence in $\C$.
Let $\sequence {z_n}$ be convergent to the following limit:
:$\ds \lim_{n \mathop \to \infty} z_n = c$
Let $\lambda \in \C$.
Then:
== Also see ==
* Sum Rule for Sequences
* Difference Rule for Sequences
* Product Rule for Sequences
* Quotient Rule for Sequences
== Sources ==
* 
* 
Category:Combination Theorems for Sequences
Combination Theorem for Sequences/Normed Division Ring

== Theorem == 
Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring.
Let $\sequence {x_n}$, $\sequence {y_n} $ be sequences in $R$.
Let $\sequence {x_n}$ and $\sequence {y_n}$ be convergent in the norm $\norm {\, \cdot \,}$ to the following limits:
:$\ds \lim_{n \mathop \to \infty} x_n = l$
:$\ds \lim_{n \mathop \to \infty} y_n = m$
Let $\lambda, \mu \in R$.
Then the following results hold:
=== Sum Rule ===
=== Difference Rule ===
=== Multiple Rule ===
=== Combined Sum Rule ===
=== Product Rule ===
=== Inverse Rule ===
=== Quotient Rule ===
Category:Combination Theorem for Sequences in Normed Division Rings
Combination Theorem for Sequences/Normed Division Ring/Combined Sum Rule

== Theorem == 
Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring.
Let $\sequence {x_n}$, $\sequence {y_n}$ be sequences in $R$.
Let $\sequence {x_n}$ and $\sequence {y_n}$ be convergent in the norm $\norm {\, \cdot \,}$ to the following limits:
:$\ds \lim_{n \mathop \to \infty} x_n = l$
:$\ds \lim_{n \mathop \to \infty} y_n = m$
Let $\lambda, \mu \in R$.
Then:
:$\sequence {\lambda x_n + \mu y_n }$ is convergent and $\ds \lim_{n \mathop \to \infty} \paren {\lambda x_n + \mu y_n} = \lambda l + \mu m$
== Proof ==
From the Multiple Rule for Sequences in Normed Division Ring, we have:
:$\ds \lim_{n \mathop \to \infty} \paren {\lambda x_n} = \lambda l$
:$\ds \lim_{n \mathop \to \infty} \paren {\mu y_n} = \mu m$
The result now follows directly from the Sum Rule for Sequences in Normed Division Ring:
:$\ds \lim_{n \mathop \to \infty} \paren {\lambda x_n + \mu y_n} = \lambda l + \mu m$
== Also see ==
* Multiple Rule for Sequences in Normed Division Ring
* Sum Rule for Sequences in Normed Division Ring
Category:Combination Theorem for Sequences in Normed Division Rings
Combination Theorem for Sequences/Normed Division Ring/Difference Rule

== Theorem == 
Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring.
Let $\sequence {x_n}$, $\sequence {y_n}$ be sequences in $R$.
Let $\sequence {x_n}$ and $\sequence {y_n}$ be convergent in the norm $\norm {\,\cdot\,}$ to the following limits:
:$\ds \lim_{n \mathop \to \infty} x_n = l$
:$\ds \lim_{n \mathop \to \infty} y_n = m$
Then:
:$\sequence {x_n - y_n}$ is convergent and $\ds \lim_{n \mathop \to \infty} \paren {x_n - y_n} = l - m$
== Proof ==
From Sum Rule for Sequences in Normed Division Ring:
:$\ds \lim_{n \mathop \to \infty} \paren {x_n + y_n} = l + m$
From Multiple Rule for Sequences in Normed Division Ring:
:$\ds \lim_{n \mathop \to \infty} \paren {-y_n} = -m$
Hence:
:$\ds \lim_{n \mathop \to \infty} \paren {x_n + \paren {-y_n} } = l + \paren {-m}$
The result follows.
== Also see ==
* Sum Rule for Sequences in Normed Division Ring
* Multiple Rule for Sequences in Normed Division Ring
Category:Combination Theorem for Sequences in Normed Division Rings
Combination Theorem for Sequences/Normed Division Ring/Inverse Rule

== Theorem == 
Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring with zero: $0$.
Let $\sequence {x_n}$ be a sequence in $R$.
Let $\sequence {x_n}$ be convergent in the norm $\norm {\, \cdot \,}$ to the following limit:
:$\ds \lim_{n \mathop \to \infty} x_n = l$
Suppose $l \ne 0$.
Then:
:$\exists k \in \N : \forall n \in \N: x_{k + n} \ne 0$
and the subsequence $\sequence { x_{k+n}^{-1} }$ is well-defined and convergent with:
:$\ds \lim_{n \mathop \to \infty} {x_{k + n} }^{-1} = l^{-1}$.
== Proof ==
Since $\sequence {x_n}$ converges to $l \ne 0$, by Sequence Converges to Within Half Limit then:
:$\exists k \in \N: \forall n \in \N: \dfrac {\norm l} 2 < \norm {x_{k + n} }$
By :
:$\forall n \in \N : x_{k + n} \ne 0$
Let $\sequence {y_n}$ be the subsequence of $\sequence {x_n}$ where $y_n = x_{k + n}$. 
By Limit of Subsequence equals Limit of Sequence, $\sequence {y_n}$ is convergent with:
:$\ds \lim_{n \mathop \to \infty} y_n = l$
It also follows that: 
:$\forall n \in \N : y_n \ne 0$
So $\sequence { {y_n}^{-1} }$ is well-defined and:
:$\sequence { {y_n}^{-1} } = \sequence { {x_{k + n} }^{-1} }$
=== Lemma ===
== Sources ==
Category:Combination Theorem for Sequences in Normed Division Rings
Combination Theorem for Sequences/Normed Division Ring/Inverse Rule/Lemma

== Theorem == 
Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring with zero: $0$.
Let $\sequence {x_n}$ be a sequence in $R$.
Let $\sequence {x_n}$ be convergent in the norm $\norm {\, \cdot \,}$ to the following limit:
:$\ds \lim_{n \mathop \to \infty} x_n = l$
Let $\sequence {y_n}$ be a subsequence of $\sequence {x_n}$.
Suppose $l \ne 0$, and for all $n: y_n \ne 0$.
Then:
:$\ds \lim_{n \mathop \to \infty} {y_n}^{-1} = l^{-1}$
== Proof ==
By Limit of Subsequence equals Limit of Sequence then $\sequence {y_n}$ is convergent with:
:$\ds \lim_{n \mathop \to \infty} y_n = l$
Let $\epsilon > 0$ be given.
Let $\epsilon' = \dfrac {\epsilon {\norm l}^2 } {2}$.
Then:
:$ \epsilon' > 0$
As $\sequence {y_n} \to l$, as $n \to \infty$, we can find $N_1$ such that:
:$\forall n > N_1: \norm {y_n - l} < \epsilon'$
As $\sequence {y_n}$ converges to $l \ne 0$, by Sequence Converges to Within Half Limit:
:$\exists N_2 \in \N: \forall n > \N_2: \dfrac {\norm l} 2 < \norm {y_n}$
or equivalently:
:$\exists N_2 \in \N: \forall n > \N_2: 1 < \dfrac {2 \norm {y_n} } {\norm l}$
Let $N = \max \set {N_1, N_2}$. 
Then $\forall n > N$:
:$(1): \quad \norm {y_n - l} < \epsilon'$
:$(2): \quad 1 < \dfrac {2 \norm {y_n} } {\norm l}$
Hence:
Hence:
:$\sequence { {y_n}^{-1} }$ is convergent with $\ds \lim_{n \mathop \to \infty} {y_n}^{-1} = l^{-1}$.
Category:Combination Theorem for Sequences in Normed Division Rings