Can we find for any real number $x$ the sequence of rationals $q\_n(x)$ with properties:


* $\lim\limits\_{n\to\infty} q\_n(x)=x$
* $q\_n(x+y)=q\_n(x)+q\_n(y)$
* $q\_n(xy)=q\_n(x)q\_n(y)$


?



Rational representation of reals
Assume $q(x)\neq 0.$ Then $q(2x)=q(x+x)=q(x)+q(x)=2q(x)$ but also $q(2x)=q(2)q(x)$ so $q\_n(2)=2$ for all $n.$ But then $2=q(2)=q(\sqrt{2}^2)=q(\sqrt{2})^2$ leading to $q(\sqrt{2})=\sqrt{2}$ which is not rational. 


For any rational $r$ one has $q\_n(r)=r$ for all $n.$ 


