Real numbers/Convergent sequences/Rules/Fact/Proof

Proof

(1). Denote the limits of the sequences by ${}x$ and ${}y$, respectively. Let ${}\epsilon >0$ be given. Due to the convergence of the first sequence, there exists for

${}\epsilon '={\frac {\epsilon }{2}}\,$

some ${}n_{0}$ such that for all ${}n\geq n_{0}$ the estimate

${}\vert {x_{n}-x}\vert \leq \epsilon '\,$

holds. In the same way there exists due to the convergence of the second sequence for ${}\epsilon '={\frac {\epsilon }{2}}$ some ${}n_{0}'$ such that for all ${}n\geq n_{0}'$ the estimate

${}\vert {y_{n}-y}\vert \leq \epsilon '\,$

holds. Set

${}N={\max {\left(n_{0},n_{0}'\right)}}\,.$

Then for all ${}n\geq N$ the estimate

${}{\begin{aligned}\vert {x_{n}+y_{n}-(x+y)}\vert &=\vert {x_{n}+y_{n}-x-y}\vert \\&=\vert {x_{n}-x+y_{n}-y}\vert \\&\leq \vert {x_{n}-x}\vert +\vert {y_{n}-y}\vert \\&\leq \epsilon '+\epsilon '\\&=\epsilon \end{aligned}}$

holds.

(2). Let ${}\epsilon >0$ be given. The convergent sequence ${}{\left(x_{n}\right)}_{n\in \mathbb {N} }$ is bounded, due to fact, and therefore there exists a ${}D>0$ such that ${}\vert {x_{n}}\vert \leq D$ for all ${}n\in \mathbb {N}$. Set ${}x:=\lim _{n\rightarrow \infty }x_{n}$ and ${}y:=\lim _{n\rightarrow \infty }y_{n}$. We put ${}C:={\max {\left(D,\vert {y}\vert \right)}}$. Because of the convergence, there are natural numbers ${}N_{1}$ and ${}N_{2}$ such that

$\vert {x_{n}-x}\vert \leq {\frac {\epsilon }{2C}}{\text{ for }}n\geq N_{1}{\text{ and }}\vert {y_{n}-y}\vert \leq {\frac {\epsilon }{2C}}{\text{ for }}n\geq N_{2}.$

These estimates hold also for all ${}n\geq N:={\max {\left(N_{1},N_{2}\right)}}$. For these numbers, the estimates

${}{\begin{aligned}\vert {x_{n}y_{n}-xy}\vert &=\vert {x_{n}y_{n}-x_{n}y+x_{n}y-xy}\vert \\&\leq \vert {x_{n}y_{n}-x_{n}y}\vert +\vert {x_{n}y-xy}\vert \\&=\vert {x_{n}}\vert \vert {y_{n}-y}\vert +\vert {y}\vert \vert {x_{n}-x}\vert \\&\leq C{\frac {\epsilon }{2C}}+C{\frac {\epsilon }{2C}}\\&=\epsilon \end{aligned}}$

hold.

For the other parts, see exercise, exercise and exercise.

To fact