Does it exist a countable set such that any sequence with value in $\mathbb{N}$ is bounded by a sequence in this set?

The question is simple, and the answer may be obvious, but it eludes me... Does it exist a countable set of sequences $S \subset \mathbb{N}^\mathbb{N}$ such that

$$\forall v \in \mathbb{N}^{\mathbb{N}}, \exists u \in S, v \leq u$$

• What is your ordering $\leq$ on $\mathbb N^{\mathbb N}$? – William Stagner Mar 1 '15 at 22:05
• The usual ordering, $$v \leq u \Leftrightarrow \forall n \in \mathbb{N}, v_n \leq u_n$$ – Tryss Mar 1 '15 at 22:06

No, by a diagonalization argument. Let $S=\{s^{(n)}:n\in\Bbb N\}$, where $s^{(n)}=\left\langle s_k^{(n)}:k\in\Bbb N\right\rangle$. For each $k\in\Bbb N$ let
$$s_k=1+\max\left\{s_k^{(n)}:n\le k\right\}\;,$$
and let $s=\langle s_k:k\in\Bbb N\rangle$. Then for each $n\in\Bbb N$ we have $s^{(n)}\not\le s$.
This is an example of a 'diagonalization' argument. Since your sequences are countable you can list them as $S_N$ for natural numbers $N$ and then construct a sequence that is greater than $S_n (N)$ for all $n$ less than or equal to $N$. Then no sequence in your set will dominate the construcTed sequence.