Ordered field/Convergent sequences/Rules/1/Fact/Proof

Proof

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.

To fact