Check if the sequence converges using Cauchy criterion I need to check if the sequence of $x_n = \dfrac1{\sqrt1} + \dfrac1{\sqrt2} ... + \dfrac1{\sqrt n}$ converges using the Cauchy criterion. 
Obviously, first I have to do is to somehow check if $|x_n - x_{n+p}| $ (where $p$ is any positive whole number) is convergent, but I have no idea even how to start. How can it be solved?

For any $n \in \mathbb{N}$ we have:
\begin{align}
x_{n^2} - x_n &= \left(\frac{1}{\sqrt{1}} +\ldots + \frac{1}{\sqrt{n^2}}
 \right) - \left(\frac{1}{\sqrt{1}} +\ldots + \frac{1}{\sqrt{n}}
 \right)\\ 
&= \frac{1}{\sqrt{n+1}} + \frac{1}{\sqrt{n+2}} + \ldots + \frac{1}{\sqrt{n^2}}\\
&\ge \underbrace{\frac{1}{\sqrt{n^2}} + \frac{1}{\sqrt{n^2}} + \ldots + \frac{1}{\sqrt{n^2}}}_{n^2 - n \text{ terms}}\\
&= \frac{n^2 - n}{\sqrt{n^2}}\\
&= \frac{n^2 - n}{n}\\
&= n - 1\\
\end{align}
Therefore, $(x_n)_{n=1}^\infty$ cannot be Cauchy since $|x_{n^2} - x_n| \ge n-1$ for all $n \in \mathbb{N}$.
We conclude that $(x_n)_{n=1}^\infty$ does not converge.
