Combination Theorem for Sequences/Normed Division Ring/Product Rule/Proof 1

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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 to the limit $\ds \lim_{n \mathop \to \infty} \paren {x_n y_n} = l m$


Proof

By Convergent Sequence in Normed Division Ring is Bounded, $\sequence {x_n}$ is bounded.

Suppose $\norm {x_n} \le K$ for $n = 1, 2, 3, \ldots$.

Let $M = \max \set {K, \norm m}$.

Then:

$\norm m \le M$

and:

$\forall n: \norm{x_n} \le M$


Let $\epsilon > 0$ be given.

Then $\dfrac \epsilon {2 M} > 0$.

As $\sequence {x_n}$ converges to $l$, we can find $N_1$ such that:

$\forall n > N_1: \norm {x_n - l} < \dfrac \epsilon {2 M}$

Similarly, for $\sequence {y_n}$ we can find $N_2$ such that:

$\forall n > N_2: \norm {y_n - m} < \dfrac \epsilon {2 M}$

Now let $N = \max \set {N_1, N_2}$.

Then if $n > N$, both the above inequalities will be true.

Thus $\forall n > N$:

\(\ds \norm {x_n y_n - l m}\) \(=\) \(\ds \norm {x_n y_n - x_n m + x_n m - l m}\)
\(\ds \) \(\le\) \(\ds \norm {x_n y_n - x_n m} + \norm {x_n m - l m}\) Axiom (N3) of norm (Triangle Inequality).
\(\ds \) \(=\) \(\ds \norm {x_n \paren {y_n - m } } + \norm {\paren {x_n - l } m}\)
\(\ds \) \(\le\) \(\ds \norm {x_n} \cdot \norm {y_n - m} + \norm {x_n - l} \cdot \norm m\) Axiom (N2) of norm (Multiplicativity).
\(\ds \) \(\le\) \(\ds M \cdot \norm {y_n - m} + \norm {x_n - l} \cdot M\) since $\sequence {x_n}$ is bounded by $M$ and $\norm m \le M$
\(\ds \) \(\le\) \(\ds M \cdot \dfrac \epsilon {2 M} + \dfrac \epsilon {2 M} \cdot M\)
\(\ds \) \(=\) \(\ds \dfrac \epsilon 2 + \dfrac \epsilon 2\)
\(\ds \) \(=\) \(\ds \epsilon\)

Hence:

$\sequence {x_n y_n}$ is convergent.

It follows that:

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

$\blacksquare$

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