A question on sequences and a inequality

Question

Given three positive sequences $\{x_n\},\{y_n\},\{z_n\}$, $x_n>0, y_n>0, z_n>0$.

It is known that, for all $n>N_0$, $|x_n-y_n|< z_n$.

Can we say that there exist constants $K_0,K_1$, independent of $n$ such that, for sufficeintly large $n$,

$$x_n<y_n + K_0 z_n$$ and

$$x_n>y_n - K_1 z_n $$

Answer 1

Yes. In fact you can make those inequalities true for all $n$. Take $K_0=1+\max \{1, |x_1-y_1|,|x_2-y_2|,...,|x_{N_0}-y_{N_0}|\}$. Can you find $K_1$ by a similar approach?

If you want those inequalities to hold just for sufficiently large $n$ then you can take $K_0=K_1=1$.

Answer 2

For sufficient large n,

$ x_n-y_n \leq |x_n-y_n|< z_n\to $\begin{align}x_n< y_n+K_0z_n ,K_0=1\end{align}

$y_n-x_n\leq |y_n-x_n|< z_n\to $\begin{align}x_n>y_n-K_1z_n ,K_1=1\end{align}