I'm a bit confused as to how to show a sequence is Cauchy by definition. For example let $a_n = 1 - \frac{2}{\sqrt{n}}$.
$\left|\left( 1 - \frac{2}{\sqrt{n}}\right) - \left( 1 - \frac{2}{\sqrt{m}}\right)\right| = \left|\frac{2}{\sqrt{m}} - \frac{2}{\sqrt{n}}\right|$
And let $m \geq n$. Then this is where I got lost in class: $|\frac{2}{\sqrt{m}} - \frac{2}{\sqrt{n}}| \leq |\frac{2}{\sqrt{n}} + \frac{2}{\sqrt{n}}|$.
Can someone explain why the last step is justified? Is it because $m \geq n \implies \frac{2}{\sqrt{m}} \leq \frac{2}{\sqrt{n}}$ and $\frac{2}{\sqrt{m}} - \frac{2}{\sqrt{n}} \leq 0 \leq \frac{2}{\sqrt{n}} + \frac{2}{\sqrt{n}}$? But the inequality may not hold when taking absolute values of both sides...