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**Alibiki** You might try one of the standard mathematical tricks: Do nothing, but do it in a smart way!

In this case one such trick is to write

$\displaystyle 0 = X_{n}Y-X_{n}Y$

and insert this in the salient difference

$\displaystyle |X_{n}Y_{n}-XY| \ .$

Invoking the Triangle inequality you can write

$\displaystyle |X_{n}Y_{n}-XY| \leq |X_{n}-X||Y| + |Y_{n}-Y||X_{n}| \ .$

If the left-hand side of this inequality is larger than some small positive number $\displaystyle \varepsilon\ ,$ then at least one of the two terms on the right-hand side of the inequality is greater than $\displaystyle \varepsilon/2\ .$