(1). Denote the limits of the sequences by
and
, respectively. Let
be given. Due to the convergence of the first sequence, there exists for
-
![{\displaystyle {}\epsilon '={\frac {\epsilon }{2}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1662e5d5610d860298747ddb4036f7a11980efac)
some
such that for all
the estimate
-
![{\displaystyle {}\vert {x_{n}-x}\vert \leq \epsilon '\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/08538bef338972a895e5a418fbee166aa9257ff6)
holds. In the same way there exists due to the convergence of the second sequence for
some
such that for all
the estimate
-
![{\displaystyle {}\vert {y_{n}-y}\vert \leq \epsilon '\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/01c5c6259068e0c14b16fe1caca0ca5f2e29d27b)
holds. Set
-
![{\displaystyle {}N={\max {\left(n_{0},n_{0}'\right)}}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/869a99ee60ec60777df597ead7824e9920b9609c)
Then for all
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7034795d756e9b740062921b326caa07c597ae70)
holds.
(2). Let
be given. The convergent sequence
is
bounded,
due to
fact,
and therefore there exists a
such that
for all
.
Set
and
.
We put
.
Because of the convergence, there are natural numbers
and
such that
-
These estimates hold also for all
.
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/49164e05998a7709f1d31c3542f09a009413afe2)
hold.
For the other parts, see
exercise,
exercise
and
exercise.