Are monotonic and continuous preferences necessarily rational? Let $\succsim$ be a strictly monotonic and continuous preference relation, and let $X=\mathbb{R}^{n}$ be the consumption set.

Is rationality of $\succsim$ implied by these conditions? 

I think transitivity is implied by continuity. However, completeness is troubling, as there are elements $x,y \in X$ that cannot be ordered with respect to $\leq$ or $\geq$, and so we cannot use monotonicity to show that $\succsim$ is complete.

I have thought of constructing a sequence $x_{n}$ with $x_{1}=x$ such that $x_{n} \to y$ and either $x_{n}\succsim x_{n+1}$ or $x_{n+1} \succsim x_{n}$. Then by transitivity and continuity we could show that $x$ and $y$ can be ordered with respect to $\succsim$, but I do not think it is possible to construct such a sequence. 

Any help would be appreciated, but please give hints and not full solutions.
Consider a preference relation in $\mathbb{R}^2$ such that $x=(x_1,x_2)\succsim (y_1,y_2)=y$ $\iff$ $x_1\geq y_1$ and $x_2\geq y_2$. 

1) You might like to argue whether this preference relation is strictly monotonic and continuous. 

2) Is the relation defined above complete?

Then, as a side dish, you might also reconsider your claim that continuity is the cause of transitivity.

Note: I just wrote this particular one with the purpose of providing a thought experiment. More in a way to challenge your understanding. I am not sure whether this example provides an answer to your question or not.