This is kind of a continuation of a recent (closed) question.

Is there an order-preserving surjective function $f:{\mathbb N}^{\mathbb N}\to [0,\infty)$ (where for $a,b\in {\mathbb N}^{\mathbb N}$ we have $a\leq b$ if $a(n) \le b(n)$ for all $n\in {\mathbb N}$)?

Thanks to Jeremy Rickard who made me aware that a previous version of this question was trivial and therefore uninteresting.