Decomposing a rational number $t \le x+y$ into the sum of two rationals, one less than real number $x$ and the other less than real number $y$

Question

Suppose $t \le x+y$, where $t$ is rational and $x,y$ are real. Must it be the case that $t=t_1+t_2$, where $t_1,t_2$ are rationals s.t. $t_1 \le x$ and $t_2 \le y$?

Answer 1

Not necessarily. Let $x=\sqrt 2$ and $y=-\sqrt 2$ and $t=0$. Clearly, $t\le x+y$. But you cannot find rational $t_1,t_2$ with $t_1+t_2=0$ and $t_1\le \sqrt 2$ and $t_2\le -\sqrt 2$. Indeed, from $t_2=-t_1$ we would conclude $t_1\le \sqrt 2\le t_1$, hence $\sqrt 2\in \Bbb Q$.