Finding a sequence of positive integers with fixed properties.

Question

Let's fix $n \in \mathbb{N} \setminus \{0\}$, $k \in \{0,\dots,n\}$.

Now consider $(M_N)_{N \in \mathbb{N}} \subseteq \mathbb{N}$ such that:

1. $\exists \,m \ge n \mid \forall \,N \ge m \quad k \le M_N \le N+k-n$;

2. $\lim_{N \to +\infty} \frac{M_N}{N}=p \in (0,1)$.

I need to find such a sequence of $M_N$. I managed to find one that fits almost all the conditions above:

$M_N:=pN+k-n \quad \forall \,N \in \mathbb{N}$ where $p \in (0,1)$.

In fact this sequence satisfies 1) and 2). However this sequence is not contained in $\mathbb{N}$.

Any idea? Thank you!