TITLE: Show that the sequence $x_n = \sqrt{n^2 + 1} - \sqrt{n + 1}$ is increasing
QUESTION [0 upvotes]: already tried to do $\frac{x_{n+1}}{x_n}<1$ or $x_{n+1}-x_n>0$ but I didn't get anything. any tips?
I want to use this to show the divergence. Only this step is missing from my demonstration

REPLY [2 votes]: An alternative approach: note that
$$
\begin{align}
x_n &=(\sqrt{n^2 + 1} - \sqrt{n+1}) \cdot \frac{\sqrt{n^2 + 1} + \sqrt{n+1}}{\sqrt{n^2 + 1} + \sqrt{n+1}}
\\ & = \frac{n^2 + 1 - (n + 1)}{\sqrt{n^2 + 1} + \sqrt{n+1}}
\\ & = \frac{n^2 - n}{\sqrt{n^2 + 1} + \sqrt{n+1}} \cdot \frac{1/n}{1/n}
\\ & = \frac{n - 1}{\sqrt{1 + \frac 1{n^2}} + \sqrt{\frac 1n + \frac 1{n^2}}}.
\end{align}
$$
From there, we can easily see that $x_n \to \infty$ as $n \to \infty$.
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