folyt

(ii)
|x_ny_n -{\alpha}{\beta}| < \epsilon  kellene

|x_ny_n -{\alpha}{\beta}| = |x_ny_n - {\alpha}y_n + {\alpha}y_n -\alpha\beta | \leq |x_ny_n - \alphay_n | + |\alphay_n - \alpha\beta | = |x_n - \alpha | |y_n| + |\alphaˇ|y_n - \beta | (konvergens sorozat, korlátos is) \leq 
\leq |x_n - \alpha | K + |\alpha||y_n - \beta | \leq K~ (|x_n - \alpha| + |y_n - \beta|) (\epsion~ := ||\Äpsilon=|| / K~) < K~ . \epsilon / K~

K~ := max{K, |\alpha|}űűhiszen \forall \epsilon~ -hoz \exists N_0 \in \mathbb{N} : |x_n - \alpha | < \epsilon~ és |y_n - \beta| < \epsilon~

(iii)
| x_n / y_n - \alpha / \beta | < \epsilon
| x_n\beta - \alphay_n / y_n\beta | < \epsilon

|x_n\beta - \alpha\beta + \alpha\beta - \alphay_nˇ/ |y_n| \beta \leq |x_n \ alpha| |\beta| + |\alpha||\beta - y_n| / |y_n||\beta| <(kellene) \epsilon

|y_n| > |\beta|/2 elég nagy n-re

< |x_n - \alpha | |\beta| + |\alpha| |y_n - \beta| / |\beta|^2 / 2


megj.:
ha x_n konvergens => \forall \epsilon > 0 \exists N_0 \in \mathbb{N} \forall n, m \geq N_0 |x_n - x_m| < \epsilon
biz.: