Can we say that, for any $n\in\mathbb {R^{+}}$ following inequality is correct?

Question

My question is so elementary. But, I don't know correct answer.

$f(n)$ and $f(n+\lambda)$ are finite functions and following fraction is convergent:

If $0≤ \lim_{n \to \infty}\frac {f(\lfloor n \rfloor)}{ f(\lfloor n +\lambda \rfloor)} <1$ for $\lambda \in \mathbb {Z^{+}}$, $ \left\{f(\lfloor n \rfloor), f(\lfloor n+\lambda \rfloor) \right\} \in \mathbb{Z^{+}}$

Here $\lambda$ is a constant.

Question: Can we say that, for any $n\in\mathbb {R^{+}}$ following inequality is correct?

$$\frac {f(\lfloor n \rfloor)}{ f(\lfloor n+\lambda \rfloor)}>\frac {f(\lfloor n+1\rfloor)}{ f(\lfloor n+1+\lambda \rfloor)}$$

I can not find a counterexample. For example $f(n)= 7^n, \lambda =1$. The rule is correct. I need a counterexample.

Thank you.

Answer 1

No, we cannot. A sequence can converge without being monotonic.