Conditions that a sequence should satisfy to be an eventually monotone sequence

Let $\{a_n\}_{n\in\mathbb{N}}$ be a sequence of real numbers such that:

1. $a_n\in[0,1]$, $\forall n\in\mathbb{N}$
2. $\lim_{n\to\infty}a_n = 0$
3. $\lim_{n\to\infty}\frac{a_{n+1}}{a_n} = 1$
4. $a_{n+1} \leq a_n$, $\forall n\in\mathbb{N}$ (decreasing sequence)
5. $\frac{a_{n+1}}{a_n} \geq \frac{a_{n}}{a_{n-1}}$ (increasing sequence)

Prove or disprove that:$$b_n = na_n \mbox{ is eventually monotone}$$so that there exists $n_0\in\mathbb{N}$ such that $\{b_n\}_{n\geq n_0}$ is a monotone sequence.

Context

Unfortunately I haven't an "analalytic form" for the sequence $a_n$... it can not be computed explicitely. Although I know that it converges to 1 and that it is bounded in $[0,1]$, and that the quotient between consecutive terms goes to $1$ (with an increasing behavior).

Idea

One idea to prove that $\{b_n\}$ is eventually monotone, is to prove that:$$\frac{b_{n+1}}{b_n} \geq 1 \mbox{ or } \frac{b_{n+1}}{b_n} \leq 1 \mbox{ $\forall n\geq n_0$}$$so that:$$\frac{(n+1)a_{n+1}}{na_n} \geq 1 \mbox{ or } \frac{(n+1)a_{n+1}}{na_n} \leq 1 \mbox{ $\forall n\geq n_0$}$$

The main problem is that I don't know the rate of convergence of $\frac{a_{n+1}}{a_n}$ to $1$, I only know that $\frac{a_{n+1}}{a_n}$ is monotone increasing.