<s>[INST] On the decreasing sequences I do not know how to prove that these sequences are decreasing:
$$
a_n=\frac{1}{\sqrt{n}(n+1)}-\frac{1}{\sqrt{n+1}(n+2)},
$$
$$
b_n=\frac{1}{n(\sqrt{n}+1)}-\frac{1}{(n+1)(\sqrt{n+1}+1)}.
$$
Thank you for all kind help and comments.
My attemption. I considered the following function
$$
f(x)=\frac{1}{\sqrt{x}(x+1)}-\frac{1}{\sqrt{x+1}(x+2)},
$$
and calculated its derivative. But its derivative is rather complicated
$$
\nabla f(x)=\frac{3x+1}{2\sqrt{x}x(x+1)^2}-\frac{3x+4}{2\sqrt{x+1}(x+1)(x+2)^2}
$$
 [/INST] Use the Mean Value Theorem to show there is an $\eta$ between $n$ and $n+1$ so that
$$
\begin{align}
a_n
&=\frac1{\sqrt{n}(n+1)}-\frac1{\sqrt{n+1}(n+2)}\\
&=\frac{\frac32\sqrt{\eta}+\frac12\frac1{\sqrt{\eta}}}{\eta(\eta+1)^2}\\
&=\frac32\frac1{\sqrt{\eta}(\eta+1)^2}+\frac12\frac1{\sqrt{\eta}^3(\eta+1)^2}
\end{align}
$$
which is obviously a decreasing function.
Likewise, show there is an $\eta$ between $n$ and $n+1$ so that
$$
\begin{align}
b_n
&=\frac1{n(\sqrt{n}+1)}-\frac1{(n+1)(\sqrt{n+1}+1)}\\
&=\frac{\frac32\sqrt{\eta}+1}{\eta^2(\sqrt{\eta}+1)^2}\\
&=\frac32\frac1{\eta^{3/2}(\sqrt{\eta}+1)^2}+\frac1{\eta^2(\sqrt{\eta}+1)^2}
\end{align}
$$
which is obviously a decreasing function.
</s>