<s>[INST] Is the sequence $\sqrt{1+\frac{1}{n^2}}$ increasing or decreasing? Is the sequence $\sqrt{1+\frac{1}{n^2}}$ increasing or decreasing?
I simplified it to $\frac{\sqrt{n^2+1}}{n}$, and I tried $a_{n+1}-a_n$ and $\frac{a_{n+1}}{a_n}$, but neither seem to work, how should I proceed?
 [/INST] Note that
$$\begin{align}
m>n\quad&\Longrightarrow\quad\frac{1}{m}<\frac{1}{n}\\\\
&\Longrightarrow\quad\left(\frac{1}{m}\right)^2<\left(\frac{1}{n}\right)^2\\\\
&\Longrightarrow\quad\frac{1}{m^2}<\frac{1}{n^2}\\\\
&\Longrightarrow\quad1+\frac{1}{m^2}<1+\frac{1}{n^2}\\\\
&\Longrightarrow\quad\sqrt{1+\frac{1}{m^2}}<\sqrt{1+\frac{1}{n^2}}\\\\
&\Longrightarrow\quad a_m<a_n
\end{align}$$
and thus the sequence is decreasing.
</s>