Is it true that if $\forall n \in {N}$ $b_n \gt a_n \land b_{n+1} - b_n \lt 0 \implies a_{n+1} - a_n \lt 0$?

Question

Let's say that I have some sequences: $b_n, a_n$.

Note that: $\forall n \in N$ $b_n \gt a_n$. Assume I'm able to say easily that $b_n$ does not increase, so can I conclude that $a_n$ does not increase either?

If I can so, what's the proof of that statement? If no, why not?

For example, I have such sequence: $a_n = \frac{1}{\sqrt{n+10}+\sqrt{n+5}}$. Now I get $b_n = \frac{1}{\sqrt{n}}$. I know that $b_n \gt a_n$ and I can easily check that $b_n$ does not increase. So now can we say that $a_n$ does not increase too?