Continuous rational and monotone preference relation implies $x\succsim0$?
If we take the definition of monotonicity to be if $x\geqq y$  then $x \succeq y$, you can simplify the proof (though it looks right). 

Note $\mathbf{0}\leq x$ for all $x\in \mathbb{R}_+^l$. So by the definition of monotonicity (essentially replacing $y$ with $\mathbf{0}$ above), $x\succeq \mathbf{0}$. I don't think continuity is required (check lexicographic preferences to see it is not necessary for the stated result).