Relationship between increasing integer sequences Suppose that $\mathcal X\cap \mathcal Y=\emptyset$, that $\mathcal X\cup \mathcal Y=\Bbb N$ and that $X(n),\;Y(n)$ are increasing surjections $\Bbb N\to \mathcal X$ respectively $\Bbb N\to \mathcal Y$. Further, suppose that there are straight line functions 
$f_{X1},f_{X2},f_{Y1},f_{Y2}$ such that $k_X=f'_{X1}=f'_{X2}$ and $k_Y=f'_{Y1}=f'_{Y2}$ so that
$$
\forall n\in \Bbb N: f_{X1}(n)\leq X(n)\leq f_{X2}(n)
$$
and
$$
\forall n\in \Bbb N: f_{Y1}(n)\leq Y(n)\leq f_{Y2}(n)
$$
Is there a determined relationship between $k_X$ and $k_Y$, like 
$k_X\cdot k_Y=k_X+k_Y$?
 A: To avoid confusion, let's write $X : \mathbb{N}\to\mathcal{X}$ and $Y : \mathbb{N}\to\mathcal{Y}$, i.e. $\mathcal{X}$ is a set, while $X$ is a function (same for $Y$ and $\mathcal{Y}$).
First, $k_X$ and $k_Y$ have to be strictly positive, because $\mathcal{X}$ and $\mathcal{Y}$ are infinite. 
Second, let's extend $f_{X1}$ and $f_{X2}$ into $\mathbb{R} \to \mathbb{R}$ functions, and let $g_X : [0,\infty) \to [X(0),\infty)$ be piecewise linear extension of $X$, that is, a function such that $\forall n \in \mathbb{N}.\ g_X(n) = X(n)$ which is linear in-between; similarly for $Y$. It follows that
$$\forall z \in [0,\infty).\ f_{X1}(z) \leq g_X(z) \leq f_{X2}(z)$$
$$\forall z \in [0,\infty).\ f_{Y1}(z) \leq g_Y(z) \leq f_{Y2}(z)$$
Observe that $X$ and $Y$ are bijections, and so are $g_X$ and $g_Y$. The above inequalities imply
$$\forall z \in [X(0),\infty).\ f_{X1}^{-1}(z) \geq g_X^{-1}(z) \geq f_{X2}^{-1}(z)$$
$$\forall z \in [Y(0),\infty).\ f_{Y1}^{-1}(z) \geq g_Y^{-1}(z) \geq f_{Y2}^{-1}(z)$$
Moreover 
$$
\frac{1}{k_X} = \lim_{z\to\infty} \frac{f_{X1}^{-1}(z)}{z} 
  \geq \lim_{z\to\infty} \frac{g_X^{-1}(z)}{z} 
  \geq \lim_{z\to\infty} \frac{f_{X_2}^{-1}(z)}{z} 
  = \frac{1}{k_X}$$
and same with $Y$, that is, $\lim_{z\to\infty}\frac{g_Y^{-1}(z)}{z} = \frac{1}{k_Y}$. Yet, because $\mathcal{X} \cup \mathcal{Y} = \mathbb{N}$, we have
$$
z = \Big|\mathcal{X} \cap[0,z]\Big| + \Big|\mathcal{Y} \cap [0,z]\Big|
  = \lfloor g_X^{-1}(z)\rfloor + \lfloor g_Y^{-1}(z)\rfloor 
  \leq g_X^{-1}(z) + g_Y^{-1}(z)
  \leq z + 2.
$$
Applying $\bullet \mapsto \lim_{z \to \infty}\frac{\bullet}{z}$ to the above inequality we arrive at
$$1 =\lim_{z\to\infty} \frac{z}{z} \leq
\lim_{z\to\infty} \frac{g_X^{-1}(z)}{z} + \lim_{z\to\infty} \frac{g_Y^{-1}(z)}{z} \leq
\lim_{z\to\infty} \frac{z+2}{z} = 1$$
which reduces to
$$\frac{1}{k_X} + \frac{1}{k_Y} = 1$$
or 
$$k_Y + k_X = k_X\cdot k_Y.$$
I hope this helps $\ddot\smile$
