Combination Theorem for Sequences

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Theorem

Sequences in Normed Division Ring

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:

$\displaystyle \lim_{n \mathop \to \infty} x_n = l$
$\displaystyle \lim_{n \mathop \to \infty} y_n = m$


Let $\lambda, \mu \in R$.


Then the following results hold:


Sum Rule

$\sequence {x_n + y_n}$ is convergent and $\displaystyle \lim_{n \mathop \to \infty} \paren {x_n + y_n} = l + m$


Difference Rule

$\sequence {x_n - y_n}$ is convergent and $\displaystyle \lim_{n \mathop \to \infty} \paren {x_n - y_n} = l - m$


Multiple Rule

$\sequence {\lambda x_n}$ is convergent and $\displaystyle \lim_{n \mathop \to \infty} \paren {\lambda x_n} = \lambda l$


Combined Sum Rule

$\sequence {\lambda x_n + \mu y_n }$ is convergent and $\displaystyle \lim_{n \mathop \to \infty} \paren {\lambda x_n + \mu y_n} = \lambda l + \mu m$


Product Rule

$\sequence {x_n y_n}$ is convergent to the limit $\displaystyle \lim_{n \mathop \to \infty} \paren {x_n y_n} = l m$


Inverse Rule

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:

$\displaystyle \lim_{n \mathop \to \infty} {x_{k + n} }^{-1} = l^{-1}$.


Quotient Rule

Suppose $m \ne 0$.

Then:

$\exists k \in \N : \forall n \in \N: y_{k + n} \ne 0$

and the sequences:

$\sequence {x_{k + n} \ {y_{k + n} }^{-1} }$ and $\sequence { {y_{k + n} }^{-1} \ x_{k + n} }$ are well-defined and convergent with:
$\displaystyle \lim_{n \mathop \to \infty} x_{k + n} \ {y_{k + n} }^{-1} = l m^{-1}$
$\displaystyle \lim_{n \mathop \to \infty} {y_{k + n} }^{-1} \ x_{k + n} = m^{-1} l$


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:

$\displaystyle \lim_{n \mathop \to \infty} x_n = l$
$\displaystyle \lim_{n \mathop \to \infty} y_n = m$


Let $\lambda, \mu \in \R$.


Then the following results hold:


Sum Rule

$\displaystyle \lim_{n \mathop \to \infty} \paren {x_n + y_n} = l + m$


Difference Rule

$\displaystyle \lim_{n \mathop \to \infty} \paren {x_n - y_n} = l - m$


Multiple Rule

$\displaystyle \lim_{n \mathop \to \infty} \paren {\lambda x_n} = \lambda l$


Combined Sum Rule

$\displaystyle \lim_{n \mathop \to \infty} \paren {\lambda x_n + \mu y_n} = \lambda l + \mu m$


Product Rule

$\displaystyle \lim_{n \mathop \to \infty} \paren {x_n y_n} = l m$


Quotient Rule

$\displaystyle \lim_{n \mathop \to \infty} \frac {x_n} {y_n} = \frac l m$

provided that $m \ne 0$.


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:

$\displaystyle \lim_{n \mathop \to \infty} z_n = c$
$\displaystyle \lim_{n \mathop \to \infty} w_n = d$


Let $\lambda, \mu \in \C$.


Then the following results hold:


Sum Rule

$\displaystyle \lim_{n \mathop \to \infty} \paren {z_n + w_n} = c + d$


Difference Rule

$\displaystyle \lim_{n \mathop \to \infty} \paren {z_n - w_n} = c - d$


Multiple Rule

$\displaystyle \lim_{n \mathop \to \infty} \paren {\lambda z_n} = \lambda c$


Combined Sum Rule

$\displaystyle \lim_{n \mathop \to \infty} \paren {\lambda z_n + \mu w_n} = \lambda c + \mu d$


Product Rule

$\displaystyle \lim_{n \mathop \to \infty} \paren {z_n w_n} = c d$


Quotient Rule

$\displaystyle \lim_{n \mathop \to \infty} \frac {z_n} {w_n} = \frac c d$

provided that $d \ne 0$.