<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] Observe that for any $n$,
\begin{align}
\sum_{k=1}^n a_k &= \sum_{k=1}^n \left(\frac1{\sqrt k(k+1)} - \frac1{\sqrt{k+1}(k+2)}\right)\\
&= \frac12 - \frac1{\sqrt{n+1}(n+2)}\\
&\stackrel{n\to\infty}\longrightarrow\frac12
\end{align}
and similarly
\begin{align}
\sum_{k=1}^n b_k &= \sum_{k=1}^n \left( \frac1{k(\sqrt k + 1)}\right) - \left( \frac1{(k+1)(\sqrt{k+1} + 1)}\right)\\
&= \frac12 - \left( \frac1{(n+1)(\sqrt{n+1} + 1)}\right)\\
&\stackrel{n\to\infty}\longrightarrow\frac12.\end{align}
Since the sequences of partial sums are monotone increasing and convergent, the sequences $\{a_n\}$ and $\{b_n\}$ are monotone decreasing.
</s>