Real numbers/Convergent sequences/Rules/Fact/Proof

Proof

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

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

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

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

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

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

holds. Set

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

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

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

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

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

{\displaystyle {}{\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.